Life Insurance Modelling: Notes for teachers. Overview

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1 Life Insurance Modelling: Notes for teachers In this mathematics activity, students model the following situation: An investment fund is set up Investors pay a set amount at the beginning of 20 years. The amount they pay depends on how likely they are to die. The fund is invested and grows with compound interest. If investors die, the fund pays beneficiaries an amount. After 20 years the survivors share the money left in the fund. Overview Students develop models that provide insight into some of the complexities that are taken into account when deciding insurance policy premiums. In particular they consider how premiums can be invested to increase the value of the fund, how the risks associated with death depend on age and gender (as well as other factors), and how their model might be developed to take account of such factors.. Students work with spreadsheets to develop their models. These models are, of necessity, simple compared to those used by actuaries and others in the insurance industries. In this sequence of lessons students work with data extracted from complex data sets associated with life expectancy. These have been simplified in the process of extraction from published sources but remain presented in the original format as far as possible. You should familiarize yourself with the data files associated with the unit before teaching the lessons. Product: Students will develop a number of spreadsheets that reflect the increasing complexity of their models and which allow calculation of insurance payouts in the event of certain events that have been insured against. The spreadsheets should allow users to input values for key variables and see how the value of the associated payouts varies. Modeling focus what students should learn: Students have opportunities to develop increasingly sophisticated models, carefully building on their increasing understanding of the real situation. They should consider how to do this by working on situations for which they making explicit assumptions, fix particular values, and then allow these to vary drawing on data and probabilities associated with risk. Suggested lesson structure and approach The unit comprises 6 lessons: Lesson 1: Exploring how savings grow and understanding the problem (savings at constant interest rate) Lesson 2: Moving on to life insurance (adding payment for death still constant probabilities) Lesson 3: Exploring variants of the model (spreadsheet changes for moving to: constant annual payments in; varying annual rates of interest from data; etc) Lesson 4: Different premiums (new word) for different people: eg males and females; calculations from life data Lesson 5: Taking account of age and discussing the modeling process 1

2 Lesson 1: Starting to explore and understand the situation Overview In this lesson students develop a spreadsheet model of the value of an insurance fund as it grows over a 20- year period with compound interest. They develop the model to consider how the amount in the fund varies because of a) the amount invested and b) the interest rate and they go on to discuss the assumptions underpinning their model. Towards the end of the lesson they are introduced to the work for the next lesson: to develop their model to include payouts for those who die and how they need to vary this to take account of a) the number of investors, b) the amount invested by each person c) amounts paid out of the fund, which in turn depends on d) the number of deaths and e) the amount paid to each of the beneficiaries of those who died. Aims/Objectives: To consider some of the important factors associated with investments and model how an investment fund might grow over time. Lesson outline: Students are given the following scenario: An investment fund is set up Investors pay a set amount at the beginning of 20 years. The amount they pay depends on how likely they are to die. The fund is invested and grows with compound interest. If investors die, the fund pays beneficiaries an amount. After 20 years the survivors share the money left in the fund. Students will probably be aware that investing money in a savings scheme attracts interest so that after one year it is worth more than the amount invested. They should also be aware of compounding that as each year passes, if interest rates stay the same throughout and the money is left in the bank, the amount in the account increases by an increasing amount. It is important that students understand that the reason why the investment increases by an increasing amount each year is because the interest is earned not only on the invested amount but also the interest earned to- date. Following a brief introduction to the purpose of the unit discuss with the class how the fund will grow if the amount in the fund is subject to simple and compound interest. Use questions such as: What happens to your money if you invest a lump sum in a bank account for a long time? What would it be worth after one year? Five years? Twenty years? What would happen to the amount by which your savings increased each year if you never took any money out of the account? Ask the students to develop a spreadsheet to show how much a lump sum investment is worth at the end of each year, for up to twenty years. If they struggle to get started, encourage them to pick an amount to invest (e.g. $50,000) and an interest rate (e.g. 2%). It is not envisaged that students should set up the investment amount and the interest rate as input variables at this stage, as doing this is planned later in the lesson. However, it may suit you and your class to do this from the outset. 2

3 Some students may need to be reminded that 2% is 2/100 or Tell the students that you want them to develop their model by allowing someone to enter any amount for the initial investment and any interest rate as input variables. They should be able to type these in and the spreadsheet should adjust automatically. A worked spreadsheet (Stage1.xls) is available to show the class. Whether or not you show it to the class at this stage will depend on your class and their experience with spreadsheets. Note that throughout this unit examples of the sorts of spreadsheets the students might produce are provided. The convention adopted is that the labels and quantities of amount that the user should be able to vary (e.g. interest rate) are coloured yellow. The spreadsheet includes a graph, which is not necessary but gives a clear visualisation of how the investment grows over time. You and your class might like to look at how the shape of the graph changes with different interest rates. You may like to discuss how the compounding gives rise to a curve and its gradient gets steeper as time increases. So that the effect of different interest rates can be seen the scaling of vertical axis of the graph has been fixed. You might need to remind students that if they want a variable to remain fixed when the formula is replicated, they should use the $ sign. (e.g. if the interest rate is in cell C2 their formula should use $C$2.) Ask the students to identify any assumptions that they made in developing their model, and to consider how realistic the model is. There are three main assumptions that underpin the modeling situation presented in this lesson: The interest is paid once a year The interest remains constant throughout the investment period No money is deposited into, or withdrawn from, the fund during the investment period. The model encompasses all the assumptions that have been made and how these have been handled mathematically as well as the spreadsheet itself. The spreadsheet plays an important role in calculating and it embodies the assumptions that have been made, but the model is more than the spreadsheet. In considering how realistic the model is, you could use questions such as the following, How often is interest likely to be paid? Is the interest likely to vary over a 20-year period? What is a realistic value for the interest rate? Under what circumstances might someone invest a lump sum and leave it invested for twenty years? Also discuss the real world situation that is being modeled. Lead the discussion towards how funds are sometimes set up by insurance companies, for example, for pension schemes. Often these are based on adding regular monthly or yearly payments a different situation that some quick students may like to model. Explore ideas of what constitutes the model and the role of the spreadsheet. Lesson 2: Developing a spreadsheet model Lesson overview In the first lesson of this unit students developed a model to explore the growth of an investment fund and discussed the process of modeling, particularly highlighting the assumptions underpinning their model. In this second lesson of the unit, they are given more details of the scenario they should model: Each member of the fund will pay in a set amount at the beginning of the period. If anyone dies, their beneficiaries will be paid out an amount. 3

4 They develop their model to reflect the more complex scenario and increase the sophistication of their spreadsheet accordingly. Towards the end of the lesson they look at, and discuss, statistics related to death rates. They will use these as a basis for further development of the model in lesson 3. Aims/Objectives: To develop a spreadsheet model of a more complex situation that allows the user to input key values to calculate final payments. Lesson outline Explain that you now want students to assume that the fund they are working on is going to be used for life insurance a mixture of saving and a payment in case of death. Each member of the fund will pay in a set amount at the beginning of the period. If anyone dies, their beneficiaries will be paid out an amount. At the end of the period, the remaining fund will be divided equally among the survivors. Note that the spreadsheet needs to allow the user to input values for: The initial amount paid in by each individual The interest rate The number of investors The amount paid out to beneficiaries if someone dies The number of deaths per year. (note: it is easiest at this stage to assume that there are a fixed number of deaths each year rather than letting this vary. You may wish to discuss this modeling assumption at this stage). Some of the assumptions that have been made are the same as those that were made in the previous lesson that investigated the growth of an investment; however, there are some additional assumptions made here. Overall, it has been assumed that: All investors invest the same amount. The interest rate is constant Interest is paid once a year No money is deposited into, or withdrawn from, the account throughout the period The same number of people die each year You could show students the spreadsheet US_Life_tables.xls or choose another set of statistics to show them. The point is that they begin to understand how to read these tables and what patterns to look for. Note that the figures in Column B (Probability of dying between ages x to x + n) are close to the mean of the male and female figures [per ] but not exactly. This is because total male and female population numbers are taken into account. Students may find it difficult to guess the number of deaths per year, or the amount paid out per death. Try to emphasize that at this stage it does not matter what values they choose: the important things are: 1. that the mathematics of the model is correct and 2. that their spreadsheet should allow the user to vary the amount paid in per person, number of investors, interest rate, amount paid out per death and number of deaths per year. Note for the discussion if the payout per death is too high, the fund could run out of money. The spreadsheet Stage2.xls provides a worked model of the scenario. 4

5 At a suitable point draw attention to the modeling process. In particular, emphasise the assumptions that underpin the models that students are developing. To stimulate discussion about the effect that the assumptions will have on outcomes ask questions such as: What is the effect of altering (for example): The interest rate? The amount of the initial investment? The number of deaths per year? In what circumstances would the fund lose money? If you managed the fund how could you ensure a greater payout for each investor at the end? Draw the students attention to the fact that the scenario is supposed to reflect real- life. Ask them what they know about life expectancy and death rates for males and females. Show them some real data, and ask them to figure out what this means. Spend a little time exploring what students notice. They may notice, for example: until the age of 82, males are more likely to die than females the death rate rises sharply in the age group 15 24, particularly for males Stimulate some discussion about the use of the data and relate this to an important assumption that the people paying into the insurance scheme are likely to behave in a similar (or the same) way that the population used to give the data did. Ask them to think about the implications of the statistics they now have and how they would need to modify their model to take account of data like this. Lesson 3: Exploring and extending the model Lesson overview The focus of this lesson is to take account of the differences between the likelihood of men and women dying and what this means in terms of the risk to the fund in terms of number of deaths per year. Students first use the data tables they have been given to calculate how likely it is that men and women will die, and then model ways to calculate what this means in terms of the number of predicted deaths per year. Aims/Objectives: To further develop the model of the situation to calculate the different contributions made by men and women based on data related to risk based on gender. Lesson outline Discuss the statistics related to age and gender. Ask the students what they know about life insurance premiums for young men and women. Do men always pay more than women? Is that fair? You could use clippings shown here, from three different web sites to start the discussion. 5

6 Students might be very aware of the fact that men frequently pay a higher premium than women, and may find it interesting that in the European Union there has been a move away from this practice recently on equity grounds. Tell the students that you want them to build into their model a way of taking into account the gender differences in terms of death rates (irrespective of their views on how fair it is!). Ask them to develop their model first to investigate the effect of: The proportion of males and females contributing to the fund. The likelihood of death for males, females and males and females combined The predicted number of deaths per year for males and females combined. You could ask the students to begin by concentrating on the risk of death for men and women, and later work out the different contributions men and women should make to the fund at the start. Eventually you will be asking the students to adjust the model so that individuals pay in a premium which is adjusted for gender and age. This is complex so we suggest that you take it step by step, slowly building the complexity. Students might begin with taking into account risks in terms of gender and what this means in terms of the predicted number of deaths per year. In lesson 4, students could then adapt the model to calculate the different contributions men and women should make to the fund if the differences are taken into account. The accompanying spreadsheet (Stage3.xls) provides a working model of the sort of thing your students should be aiming towards. This spreadsheet uses figures based on those in a cut down version of the US_Life_tables spreadsheet you showed students at the end of the previous lesson. The name of the file used is US_Life_tables- simplified. It has used figures from for the age groups between 20 and 50, which have been averaged for both genders. The overall death rate uses these figures in proportion to the gender balance in the fund. (i.e. (male risk*% males+female risk*% females)/100). The model uses the mean of the male and female death rates from ages 20 to 50 to predict numbers of deaths. It ignores the fact that the total population decreases from to just under It also assumes a) that total populations of males and females are equal b) contributors to the fund are aged between 20 and 49. 6

7 Note that it is not important for modeling purposes, for students to use real death rates but that the experience of using empirical data is an important one in learning about how insurance companies work. What is important is that The model is mathematically sound The students use the model to explore the effects of the two new variables on the fund. Students may worry about, for example, deaths. The example model provided does not round numbers up or down, but this is something your students might like to build into their model. You may wish to indicate that this way of proceeding, that is, building the complexity of the model step- by- step is a good way to approach complex modeling. It means that not only does the model become increasingly more useful as a model of reality but also means that you can keep a close check on the accuracy of the mathematics as it becomes increasingly complex. You can highlight this by referring students to the modeling cycle and emphasising that in this unit they have had opportunities to cycle round this a number of times as they have developed their model to become more and more representative of reality. Lesson 4: Different contributions for males and females. Introducing age as a variable. Lesson overview This lesson continues to build on the idea that death rates for males tends to be higher than for females. In the previous lesson students adapted their model to take into account the different risks for men and women and to calculate the predicted number of deaths per year, depending on the gender balance within the fund and the number of people contributing to the fund. In this lesson, they further develop the model to calculate the different amounts male and female contributors to the fund should make, based on the risk they present to the fund. Towards the end of the lesson they further develop their model to take age into account (if time allows if Aims/Objectives: To consider how to make the model for life insurance more sophisticated and the spreadsheet more efficient To work through iterative cycles of model improvement in developing a more and more sophisticated model. Lesson outline Tell students that you would now like them to develop their model further to take into account men and women paying different premiums. For example, you could suggest that they assume men pay in a given amount (e.g. 1,000). Figure out what women should pay if they are less likely to die. The model thus far has assumed that all contributors to the fund make the same financial contribution. The model now needs to assume that male contributions should be more related to the risk they represent to the fund and similarly for females. The approach suggested here (to fix the male contribution) is one approach but your students may choose to take a different approach. To assist students who are having difficulty in deciding how to vary premium payments depending on a person s gender you might suggest they look at the sample file: four.xls. This shows one way of developing the model. 7

8 They will have to look carefully at the formulae in the spreadsheet cells and think what in reality the cell references represent so as to inform the development of their own spreadsheet. For students having difficulty using real data you may find it helpful to ask them to consider the special case of when men are twice as likely as women to die. (In this case it would be expected that men should contribute twice as much as women). Hold a discussion with the class about the data. Show them the simplified life tables data (US_Life_tables- grouped ages), which is taken from the life tables data seen previously. Emphasize that a simplification has been made to make the modeling easier. Ask them to look carefully at the data across and down and ask questions such as: What does this data say about risk of death for different groups of people? Remind students that their model already assumes that females pay in less than males. Ask them how else they might change the contributions made by different groups based on age groupings. The data spreadsheet, US_Life_tables- grouped ages), now includes calculations for mean death rates for six different age groups (20 24, etc). It is important that students are aware that the data gives a snap- shot of what happened in the US on a particular year. The point of this discussion is to introduce age as a variable that should be taken into account in the model. Tell the students that you would like them to add age as a variable into their model and take this into account in their spreadsheet. You might like to use some of the following questions as prompts: How many age groups should you have? Does it make sense to collapse two groups (such as and 25 29) into one group? What are the advantages and disadvantages of doing so? Which age groups should you exclude altogether? Students may decide to simplify their spreadsheet by grouping the given age groups in some way. This has been done in the example file Stage5.xls, which you might like to show them. In this spreadsheet, different contribution rates are calculated for each of six groups: three age groups for males and three for females. Each has a different risk factor, which is reflected in the contribution they make. The contributions are calculated by giving the middle male group a fixed contribution (in the example this is $1000) and all other contributions are calculated in proportion to this one. The model assumes that: The total male/female members of the fund are divided into three equal age groups. There are clearly other ways to make these calculations and you could encourage confident students to think about different methods. Lesson 5: Further taking account of age and reflecting on modelling Lesson overview In the first part of the lesson students further modify their model and their spreadsheet to take account of the risk of death by age in addition to gender. They then take a step back to think about the modeling process. 8

9 Aims/Objectives: To build an increasingly complex model of the scenario and spreadsheet that calculates what happens when age is taken into account. Lesson outline Ask the students to continue to work on their model to take into account age as well as gender. This last level of sophistication was introduced towards the end of the previous lesson, but if not, refer to the notes for lesson 4 to introduce the ideas. Reflect with the class on the modeling process and the usefulness of the model You could stimulate discussion with questions such as: What was the real world situation we modeled? How real was the model to start with? How did you make your model more real over time? How real is your model now? Also ask students about assumptions using questions such as: What are the important assumptions that underpin your final model? In which ways are these assumptions realistic? What are the benefits of making these assumptions? [To create a good enough model]. Ask students to consider the other assumptions made in creating the model and list them for everyone to see. Which assumptions are you most/least comfortable with? Why? In this last discussion of the unit, the emphasis is on the modeling process rather than the details of the model. The following important issues should come out of your discussion with students: The method used in this unit to develop a complex model, by adding an extra variable at each step, is a useful approach and can be used in a variety of situations not just when working with a spreadsheet. Decisions, assumptions and compromises are always made in modelling so as to eventually reach a best case that you can work with. Models, even when a number of quite drastic simplifying assumptions have been made can be good enough for getting a feel for a situation and certainly provide useful insights. Discuss how a model similar to this might be used in other situations. Ask students how certain aspects of this model might be applicable in other situations?[e.g. When modeling any growth of a population] 9