# 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

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

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