Retirement Problem Week Eleven fall 2023

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Toronto Metropolitan University *

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502

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Finance

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Jan 9, 2024

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Retirement Problems Problem One: You are a 25 year old male. You wish to retire at age 55. a) What is the probability of your living to age 55? b) Assuming that you live to age 55, what is your life expectancy at age 55? c) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 25%. How many years will you need your income to last during retirement? d) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 10%. How many years will you need your income to last during retirement? e) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 5%. How many years will you need your income to last during retirement? f) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 10%. How many years will you need your income to last during retirement? g) Assuming you reach 55, you want the probability of outliving your money to be zero. How many years will you need your income to last during retirement? Problem Two: You are a 25 year old female. You wish to retire at age 55. a) What is the probability of your living to age 55? b) Assuming that you live to age 55, what is your life expectancy at age 55? c) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 25%. How many years will you need your income to last during retirement? d) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 10%. How many years will you need your income to last during retirement? e) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 5%. How many years will you need your income to last during retirement? f) Assuming you reach 55, you wish to minimize the probability of running out of money during retirement, therefore you want the probability of outliving your money to be less than 10%. How many years will you need your income to last during retirement? g) Assuming you reach 55, you want the probability of outliving your money to be zero. How many years will you need your income to last during retirement? h)

Problem Three: You are a 55-year male. You wish to have an income of $90,000 per year in retirement and the payments are made at the end of the year. Fill in the table below and calculate the amount of money you will need when you retire. 4% 5% 6% 7% 8% 31 37 42 44 48 1000 Problem Four: You are a 55-year male who just retired. You wish to have an income of $90,000 during your first year of retirement. The annual income will be paid at the end of each. You believe that inflation is likely to be 3% per year during retirement. You want the payments after the initial payment to increase by the expected rate of inflation. Fill in the table below and calculate the amount of money you will need when you retire. 4% 5% 6% 7% 8% 31 37 42 44 48 1000

Problem Five: You are a 55-year female. You wish to have an income of $90,000 per year in retirement and the payments are made at the end of the year. Fill in the table below and calculate the amount of money you will need when you retire. 4% 5% 6% 7% 8% 34 40 44 46 51 1000 Problem Six: You are a 55-year female who just retired. You wish to have an income of $90,000 during your first year of retirement. The annual income will be paid at the end of each. You believe that inflation is likely to be 3% per year during retirement. You want the payments after the initial payment to increase by the expected rate of inflation. Fill in the table below and calculate the amount of money you will need when you retire. 4% 5% 6% 7% 8% 34 40 44 46 51 1000

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Problem Seven: Your mom is retiring today. She has worked for the same company for 35 years. Her salary for the past 5 years was: 90,000; 92,000; 94,500; 97,000; and 100,000. She receives an annual pension income of 2% per year of service, of her average earnings in her last 5 years of employment. a) What is your mom’s annual pension? b) Assuming that your mom lives for 35 years after retirement what is the amount of money needed to fund her future pension payments on the day she retires. The pension plan earns 7% during your mom’s retirement. Assume the payments are made at the beginning of the year. c) For different expected # of years lived during retirement and different rates of return determine the present value required to fund your mom’s pension. 4% 5% 6% 7% 25 35 45 55 d) Assume that your mom’s pension is indexed to inflation. Assume that inflation is 3% per year during retirement. If the pension earns 7% during retirement what is the amount of money needed to fund your mom’s pension assuming she lives for 35 years in retirement? Assume the payments are made at the beginning of the year. e) Assuming that your mom’s pension is indexed and that inflation is 3% during retirement, fill in the table below. For different expected # of years lived during retirement and different rates of return determine the present value required to fund your mom’s pension. 4% 5% 6% 7% 25 35 45 55 f) Your aunt (your mom’s twin) also is retiring tomorrow. She set up an RRSP when she started working. She has the same amount of money in her RRSP that your mom’s company has in its pension plan to fund your mom’s pension. How much did your aunt have to set aside at the beginning of each year, to fund her RRSP assuming that she earns 7% per year. g) Your aunt wants to receive an indexed payment during retirement. Inflation is expected to be 3%. Therefore, she will have the same amount of money as the pension plan has in part d. Now how much did your aunt need to set aside each year, assuming that she earns 7% per year. h) Suppose that your mom’s twin waited for a number of years (5 years, 10 years, 15 years, 20 years) before she started to save for retirement how much would she have to invest each year?

Problem Eight: Your mom is retiring today. She has worked for the same company for 30 years. Her salary for the past 5 years was: 120,000; 132,000; 138,000; 145,000; and 150,000. She receives an annual pension income of 2% per year of service, of her average earnings in her last 5 years of employment. a) What is your mom’s annual pension? b) Assuming that your mom lives for 25 years after retirement what is the amount of money needed to fund her future pension payments on the day she retires. The pension plan earns 5% during your mom’s retirement. Assume the payments are made at the beginning of the year. c) For different expected # of years lived during retirement and different rates of return determine the present value required to fund your mom’s pension. 4% 5% 6% 7% 25 35 45 55 d) Assume that your mom’s pension is indexed to inflation. Assume that inflation is 3% per year during retirement. If the pension earns 5% during retirement what is the amount of money needed to fund your mom’s pension assuming she lives for 25 years in retirement? Assume the payments are made at the beginning of the year. e) Assuming that your mom’s pension is indexed and that inflation is 3% during retirement, fill in the table below. For different expected # of years lived during retirement and different rates of return determine the present value required to fund your mom’s pension. 4% 5% 6% 7% 25 35 45 55 f) Your aunt (your mom’s twin) also is retiring tomorrow. She set up an RRSP when she started working. She has the same amount of money in her RRSP that your mom’s company has in its pension plan to fund your mom’s pension. How much did your aunt have to set aside at the beginning of each year, to fund her RRSP assuming that she earns 5% per year. g) Your aunt wants to receive an indexed payment during retirement. Inflation is expected to be 3%. Therefore, she will have the same amount of money as the pension plan has in part d. Now how much did your aunt need to set aside at the beginning each year, assuming that she earns 5% per year. h) Suppose that your mom’s twin waited for a number of years (5 years, 10 years, 15 years, 20 years) before she started to save for retirement how much would she have to invest each year?

i) Suppose that your aunt was planning on earning 7% per year for prior to retirement. She started saving for retirement when she started work. She planned on making contributions to her RRSP for the 30 years she was working. She would make the contributions at the beginning of each year. During the first year her effective annual rate of return of -20%. To meet her goal by how much does her annual contribution have to increase by? j) Suppose that your aunt was planning on earning 7% per year for prior to retirement. She started saving for retirement when she started work. She planned on making contributions to her RRSP for the 30 years she was working. She would make the contributions at the beginning of each year. During the first 10 years of contributions her effective annual rate of return of 2%. To meet her goal by how much does her contribution have to increase by?

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Problem Nine: Suppose there are two brothers Samuel and Martin. Samuel contributes $6,000 per year to a RRSP. He makes his contributions at the beginning of the year. Samuel makes a total of 45 contributions. He earns 8% compounded annually. In 45 years, Samuel converts his RRSP into a 30 year annuity. The annuity earns 5% per year. The annuity makes annual payments at the beginning of every year. a) How much does Samuel have when he retires? b) How much is the annual annuity that Samuel receives? c) Calculate the amount of the annuity payment that Samuel receives assuming that he earns 4%, 5%, 6%, 7% or 8% before retirement and the annuity earns 1%, 2%, 3%, 4%, 5% or 6% during retirement. Problem Ten: Suppose there are two brothers Samuel and Martin. Martin contributes $6,000 per year to a TFSA. He makes his contributions at the beginning of the year. Martin makes a total of 45 contributions. He earns 8% compounded annually. In 45 years, Martin starts to withdraw money from his TFSA. Assume the TFSA continues to earn 8% compounded annually. The withdrawals are made at the beginning of each year. a) How much does Martin have when he retires? b) How much is the annual annuity that Martin receives? c) Calculate the amount of the annuity payment that Martin receives assuming that he earns 4%, 5%, 6%, 7% or 8% before retirement and he earns only 3%, 4%, 5%, 6%, 7% or 8% during retirement. Problem Eleven: Your grandmother just purchased a $100,000 5-year annuity. The annuity will make monthly payments of $1,750.72. The payments are made at the end of every month. What is the effective annual rate of return that your grandmother is earning? Problem Twelve: Your grandfather just purchased a $100,000 20-year annuity. The annuity will make monthly payments of $558.76. The payments are made at the end of every month. What is the effective annual rate of return that your grandfather is earning?