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STATS 3G03: Assignment 4

You are allowed/encouraged to use Excel or R provided it is all your own work from scratch (and will be required to for question 6). When using software, explain in your answers what calculation is being done, and provide a sample calculation. Upload a copy of the spreadsheet or code to the dropbox along with the assignment solutions. Using someone else’s code for your own credit is an act of academic dishonesty, and both the user and supplier are charged.

DO NOT SHARE YOUR CODE WITH ANYONE OR YOU MAY BE CHARGED.

1. You are given the following select and ultimate table.

A client of age 73 was provided 6 years of insurance 2 years ago. The insurance pays $200,000 at the end of the year of death if they die before age 77. They would now like to purchase additional insurance (without going through the underwriting process again) to extend that insurance benefit to provide that same coverage if they die before age 80. What is the expected present value of that additional coverage and its actuarial notation? The annual effective interest rate is 5%.

2. The one-year probablility of the failure of a newly purchased machine is given by  for integer values of x < 10, the age of the machine. The CONTINU-OUSLY COMPOUNDING interest rate is δ = 9%.

a) Compute the EPV of a discrete insurance benefit on a new machine (age x = 0) that pays $60,000 at the end of the year in which the machine fails.

b) Using your answer from part (a), and the relations determined in the notes, find the EPV of the same insurance benefit if it pays at the end of the month in which the failure occurs. You will assume failure has a UDD during each year. No credit will be given for sums involving indiviudal monthly probabilities.

c) Recompute the answer to part (b) using the other approximation that was determined in the notes (no credit will be given for sums involving indiviudal monthly probabilities).

3. Depicted in the picture below is the graph of the size of the continuous insurance benefits for a policy of term 20 years. Determine how the shape of the benefit can be constructed using policies involving variations of insurance benefits that are constant (like  for instance) and linearly increasing (like ). Provide two different combinations:

i) a combination that can include the use of deferred insurance, and

ii) a combination without any deferred insurance policies.

You can include “minus” any number of term insurance benefits.

For number (ii), I recommend starting from the right side of the picture and working left.

4. Similar to the previous question, find some combination of term insurance involving fixed payouts, and increasing payouts (IA) in order to provide a 25-year-old insurance that pays $300,000 at the end of the year of death if death occurs before the age of 40, but for which the benefit then decreases linearly down to a benefit of zero after turning 60. The benefit then remains at zero thereafter. As in the previous question, answer with and without using deferred insurance.

5. Find the EPV of a whole life insurance policy that pays an amount at the time of death that begins at $150,000 at t = 0, but then decreases exponentially with time, such that it is down to $120,000 after 5 years (it continues to decrease at that same constant exponential rate afer that. There is no change in shape at t = 5). Use a force of mortality that is a constant 6%, and a constant force of interest of 3%.

6. The following life table is provided in a separate Excel sheet.

The annual effective interest rate is 7%. Create 3 new columns in the spreadsheet:

a) The column for dx

b) A column for the EPV of $500 per year paid at the beginning of each year that they are alive USING RECURSION.

c) A column for the EPV pf $500 per year paid at the end of each year that they are alive USING THE VALUES IN THE COLUMN FROM PART (b)

d) Confirm your results by find the EPV of a whole life annuity of $500 per year paid by an 81-year-old if the payments are made at the end of each year, but this time by calculating it directly as a sum of several EPVs rather than using recursion.