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6.06 Perpetuities

Worksheet
Perpetuities
1

The Principal of a high school wants to start a scholarship where the top student each year receives a \$4000 prize. The Principal wants to invest money at the beginning of the year so that the interest earned by the end of the year covers this prize. If the interest on the initial investment averages 4\% per annum, compounded annually, how much should his initial investment be?

2

Anousheh wants to make an ongoing donation of \$4300 to World Vision each year from the interest earned on an investment. If the interest on the initial investment averages 2.15\% per annum, compounded annually, how much should his initial investment be?

3

Asghar invests his superannuation payout of \$220\,000 in a perpetuity that pays 6\% per annum, compounding annually. What is the size of the annual payment he will receive?

4

Samih has just won a \$5\,000\,000 jackpot and decides to invest it in a perpetuity that pays 3\% per annum, compounding monthly. What is the size of the monthly payment he will receive?

5

Farzad invests his workers' compensation payout of \$2\,760\,000 in a perpetuity that pays 2.85\% per annum, compounding quarterly. What is the size of the quarterly payment he will receive?

6

Jenny receives \$600\,000 from an inheritance and wishes to invest the money so that her interest payments cover her monthly living expenses of \$1500 per month. The investment will be compounded monthly.

Find the annual interest rate, r, that she will need for this investment.

7

Glen won the lotto and has invested the money in a perpetuity paying 8\% per annum compounded quarterly. He is able to pay himself \$25\,000 every quarter from his account without using any of the principal.

How much money did he win?

Perpetuities with recurrence relations
8

\$16\,000 is invested in a perpetuity at 3\% per annum, compounded annually. A constant amount is withdrawn from the account at the end of each year. This perpetuity can be defined recursively by A_{n + 1} = a A_n - b, A_0 = c, where A_{n + 1} is the amount remaining in the account after n + 1 years.

Find the values of a, b and c.

9

\$30\,000 is invested in a perpetuity at 6\% per annum, compounded monthly. A constant amount is withdrawn from the account at the end of each month. This perpetuity can be defined recursively by A_{n + 1} = a \times A_n - b, A_0 = c, where A_{n + 1} is the amount remaining in the account after n + 1 months.

Find the values of a, b and c.

10

\$400\,000 is invested in an account at 4\% per annum, compounded annually. \$33\,000 is withdrawn from the account at the end of each year.

a

Write a recursive rule that gives the balance of the account, A_n, at the end of year n.

b

Determine the amount remaining in the account at the end of 13 years.

c

Does this investment represent a perpetuity? Explain your answer.

d

What amount should be withdrawn each year so this investment is a perpetuity?

11

\$300\,000 is invested in an account at 6\% per annum, compounded annually. \$18\,000 is withdrawn from the account at the end of each year.

a

Write a recursive rule that gives the balance of the account, A_n, at the end of year n.

b

Determine the amount remaining in the account at the end of 12 years.

c

Does this investment represent a perpetuity? Explain your answer.

12

\$150\,000 is invested in an account at 7.2\% per annum, compounded monthly. \$1200 is withdrawn from the account at the end of each month.

a

Write a recursive rule that gives the balance of the account, A_n, at the end of month n.

b

Determine the amount remaining in the account at the end of 29 months.

c

Does this investment represent a perpetuity? Explain your answer.

d

What amount should be withdrawn each month so this investment is a perpetuity?

13

Kathleen sets up a perpetuity with \$200\,000 invested at 6.1\% per annum, compounded annually. At the end of each year she donates a constant amount to one of the local charities in her area.

a

How much does she donate each year if she wishes to be able to do so indefinitely?

b

After a few years she plans a trip of a lifetime for her and her family and withdraws \$90\,000 from this account. How much will she now have available to donate each year to a local charity?

14

Xavier invests \$350\,000 in a perpetuity earning 9.6\% per annum compounded monthly and wishes to make monthly payments to the Cancer Council for research.

a

How much does he pay the Cancer Council each month if he wishes to be able to do so indefinitely?

b

After 3 years Xavier experiences some financial difficulty and needs to withdraw \$3600 each month from this account to pay for living expenses and stops donating money to the Cancer Council. Write a recursive rule that gives the balance of the account, A_n, at the end of the nth month after this time.

c

Use the sequence facility on your calculator to determine at the end of which month the money will have run out.

15

Hermione invests her superannuation payout of \$500\,000 into a perpetuity that will provide a monthly income without using any of the initial investment. If the interest rate of the perpetuity is 9\% per annum compounded annually, what monthly payment will Hermione receive?

16

A university mathematics faculty has \$30\,000 to invest. It intends to award an annual mathematics prize of \$1500 with the interest compounded per annum from investing this money in a perpetuity. Calculate the interest rate required.

17

Lachlan wins \$500\,000 in a lottery and invests the amount in a perpetuity that pays 4.85\% per annum interest. From the perpetuity he will receive regular monthly payments.

a

Find the amount of his monthly payment.

b

How much does he receive in total over 6 years?

c

Find the balance of the account after the 6 years.

18

Sonia inherits \$265\,000 from her grandparents and invests the amount in a perpetuity that pays 3.95\% per annum interest. From the perpetuity she will receive regular quarterly payments.

a

Find the amount of her quarterly payment.

b

How much does she receive in total over 14 years?

c

Find the balance of the account after the 14 years.

19

Will Bates donates five million dollars to a charity organisation. The organisation invests the amount in a perpetuity that pays 5.05\% per annum interest. From the perpetuity, the organisation will receive regular monthly payments which will fund their charity work.

a

Find the monthly payment that the charity receives.

b

How much does the charity receive in total over 10 years?

20

A former student, who is now a professional footballer, donates a perpetual trophy and a cash prize for the fairest and best footballer at their high school. The trophy costs \$25 to be engraved each year and the prize is \$450 cash. The perpetuity is invested in an account which pays 4.5\% per annum interest.

a

Find the amount donated to the school.

b

If engraving costs rise to \$30 per year and the value of the cash prize is increased to \$500 per year, how much extra money will need to be invested in the perpetuity to ensure the balance remains stable?

21

Warren Tuffett donates six million dollars to a charity organisation. The organisation invests the amount in a perpetuity that pays 5.05\% per annum interest compounded monthly. From the perpetuity the organisation will receive a yearly payment which will fund their charity work.

a

Calculate the effective annual interest rate for the perpetuity account as a percentage to two decimal places.

b

Calculate the amount of the yearly payment the charity receives.

c

How much does the charity receive in payments in total over 7 years?

22

A former student, who is now a professional actor, donates a sum of money to provide an annual trophy and a cash prize for the best drama student at their high school. The total value of the trophy and prize is to be \$5000 and it is presented at the end of each school year. The initial donation is invested in a perpetuity account which pays 4.25\% per annum interest compounded monthly.

a

Calculate the effective interest rate for the perpetuity account, to two decimal places.

b

Hence or otherwise, determine the amount of money that the former student donated to the school.

23

A professional artist, donates a sum of money to provide an annual trophy and a cash prize for the best art student at a high school. The total value of the trophy and prize is to be \$3500 and it is presented at the end of each school year. The initial donation is invested in a perpetuity account which pays 3.85\% per annum interest compounded weekly.

a

Calculate the effective interest rate for the perpetuity account to two decimapl places. Assume there are 52 weeks in a year.

b

Hence or otherwise, determine the amount of money donated to the school.

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Outcomes

ACMGM100

with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount

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