Unformatted text preview: MATH 471: ACTUARIAL THEORY I FALL 2010: CLASS PROJECT INFO (Assigned October 18, Due December 8) This class project utilizes many of the concepts and techniques that we have discussed in MATH 471. Specifically, this project will involve constructing a life table based on birth cohort data. The life table will then be used to answer some additional questions. ______________________________________________________________________________ PRELIMINARIES: Students will work on this project in groups of either four or five students. That is, no group can consist of less than four students or more than five students. I will set aside time in class to give all students a chance to partner up. It is YOUR responsibility to find other students with whom to be in a group. Pick a member of the group to be your representative, who will be responsible for any communication with me. The representative must identify the composition of his/her group by 5:00pm on Wednesday, October 27, via e mail to me at [email protected] Any students who are not identified in a group after the deadline (5:00pm, October 27) will automatically lose three points on this project, NO EXCEPTIONS! SITUATION: Your group is to analyze birth cohort data (values of lx for ages x = 18, 19, ..., 110). Using this birth cohort data: (i) construct a life table, and (ii) use it to answer some additional questions. DATA: Shortly after receiving the email identifying the members of each group, I will email each group representative a Microsoft Excel spreadsheet containing the birth cohort data. Each group will receive different lxs than the other groups. 1 SPECIFIC ISSUES: This project can be split into two parts: PART I: CONSTRUCTION OF THE LIFE TABLE. Use Microsoft Excel and construct a life table based on the lxs provided for x = 18, 19, ..., 110. In addition to columns for x and lx, the life table should include columns for the following at each age x: 1) The probability that x survives to age (x + 1). 2) The probability that x dies prior to age (x + 1). 3) The expected number of deaths between ages x and (x + 1). 4) The curtate expectation of life for (x). 5) The APV of a discrete whole life insurance of 1 on (x) at interest rate i. 6) 2Ax at interest rate i. 7) The APV of a whole life annuitydue of 1 per year on (x) at interest rate i. 8) The APV of a 5year pure endowment of 1 on (x) at interest rate i. 9) The APV of a 10year pure endowment of 1 on (x) at interest rate i. 10) The APV of a 20year pure endowment of 1 on (x) at interest rate i. 11) The APV of a whole life insurance of 1 on (x) where the benefit is payable at the end of the mth of a year of death. Assume deaths are uniformly distributed within each year of age. 12) The APV of a whole life annuitydue on (x) with benefits of 1/m payable at the beginning of each mth of a year. Assume deaths are uniformly distributed within each year of age. Notes: A) In order to calculate 5) 12) at any annual effective interest rate i, you will need a cell in the Excel spreadsheet in which the user can enter a value for i. Also include cells for the discount factor v and the annual effective discount rate d based on the value of i. B) In order to calculate 11) 12), you will need to have a cell in the Excel spreadsheet in which the user can enter a value for m. You should also have additional cells which provide values for both the nominal rates of interest and discount (i(m), d(m)). 2 PART II: ADDITIONAL QUESTIONS. Use the life table you constructed in Part I to answer the following questions: 1) Calculate the 10year temporary curtate life expectancy for (25). 2) Assume i = 0.05. (a) Calculate the APV of a 20year term insurance of 100,000 on (20) with benefit payable at the end of the year of death. (b) Calculate the variance of Z, where Z is the present value random variable for the insurance in (a). 3) Assume i = 0.05. (a) Calculate the APV of a 20year temporary life annuitydue of 1000 per year on (20). (b) Calculate the variance of Y, where Y is the present value random variable for the annuity in (a). (c) Assuming a uniform distribution of deaths within each year of age, calculate the APV of a 20year temporary life annuitydue of 1000 per year on (20), payable on a monthly basis (actual payments are 1000/12 per month). 4) For a special fully discrete 35payment whole life insurance on (30): (i) The death benefit is 1 for the first 20 years and is 5 thereafter. (ii) The initial benefit premium paid during each of the first 20 years is one fifth of the benefit premium paid during each of the subsequent 15 years. (iii) i = 0.04 Calculate the initial annual benefit premium. 5) For a fully discrete whole life insurance of 100,000 on each of 10,000 lives age 60: (i) (ii) (iii) (iv) The future lifetimes are independent. Mortality for each life follows your life table. i = 0.06 is the premium for each insurance of 100,000. Using the normal approximation, calculate such that the probability of a positive loss is 1%. 3 GRADING: I expect each group to turn in to me: (i) the Microsoft Excel spreadsheet containing the life table, and (ii) the solutions to the five questions based on the life table. This project will be graded out of 25 points. 10 points will be available with respect to the correctness your life table. 12 points will be available with respect to the correctness of your solutions to the five questions. 3 points will be available with respect to the "professionalism" of all that you submit. For example, is your life table organized (columns clearly labeled, headings clearly identifiable, etc.) Are your solutions to the five questions complete (solutions are outlined stepbystep, neat with no crossouts, messy erasures, etc.), and organized (problems numbered, all work shown in the solutions, answers clearly indicated)? Basically, take the time to make sure that everything is neatly and logically laid out for the reader (me). Email your Excel file with the life table to [email protected] Be sure to include "MATH 471" and "Class Project Life Table" in the subject line of the email and make sure each group member's name is listed in the spreadsheet itself. All class project materials (Excel file and solutions to the five questions) are due at the beginning of the group representative's lecture section on December 8, 2010. Furthermore: 1) Projects that are submitted after the beginning of the group representative's lecture section on December 8, 2010, will NOT be accepted and will be assigned a grade of ZERO. 2) Any projects submitted by a group that does not contain either four or five students will NOT be accepted and will be assigned a grade of ZERO. 3) If your group finishes the project early, you may turn it in to me prior to December 8, 2010. CLOSING COMMENTS: I will NOT answer any questions regarding your analysis. I WILL answer questions involving clarification of the guidelines that I have outlined in this document. 4 ...
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 Spring '08
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 Math, Actuarial Science, APV, Life table

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