# 6.07 Annuities

Worksheet
Financial application
1

Iain invests \$190\,000 at a rate of 7\% per annum compounded annually. At the end of each year he withdraws \$14\,300 from the investment after the interest is paid and the balance is reinvested in the account.

a

Complete the given table, leaving out the unknown variable.

b

Use the financial solver on your CAS calculator to determine which year the annuity will run out.

2

Carl invests \$190\,000 at a rate of 12\% per annum compounded monthly. At the end of each month he withdraws \$3900 from the investment after the interest is paid and the balance is reinvested in the account.

a

Complete the given table, leaving out the unknown variable.

b

Use the financial solver on your CAS calculator to find the month during which the annuity will run out.

3

Avril invests \$190\,000 at a rate of 7\% per annum compounded annually. Avril's wants to find out what her annual withdrawal should be if she wants the investment to last 25 years. a Complete the given table, leaving out the unknown variable. b Use the financial solver on your CAS calculator to determine the amount of the annual withdrawal. 4 Victoria invests \$190\,000 at a rate of 12\% per annum compounded monthly.

Victoria wants to determine what her equal monthly withdrawal should be if she wants the investment to last 20 years.

a

Complete the given table, leaving out the unknown variable.

b

Use the financial solver on your CAS calculator to determine the amount of the monthly withdrawal.

5

Katrina has \$150\,000 to invest. She wishes to withdraw \$1400 each month after the interest is paid. The interest is compounded monthly. Katrina wants to determine what annual interest rate she needs if she wants her investment to last 30 years.

a

Complete the given table, leaving out the unknown variable.

b

Use the financial solver on your CAS calculator to determine the amount of the annual interest rate. Round your answer to two decimal places.

6

Hannah invests \$190\,000 at a rate of 16\% per annum compounded quarterly. Hannah wants to know what her quarterly withdrawal should be if she wants the investment to last 30 years. a Complete the given table, leaving out the unknown variable. b Use the financial solver on your CAS calculator to determine the amount of the monthly withdrawal. 7 Fiona opens an annuity with \$90\,000 as her initial investment. She earns 5.76\% interest per annum compounded monthly and makes an annual withdrawal of \$7000 at the end of each year which she uses to go on a holiday. a Complete the given table, leaving out the unknown variable. b Use the financial solver on your CAS calculator to determine how many years the annuity will last. Round your answer to the nearest year. c Fiona would like to work out the interest rate she would need to make the annuity to last 40 years instead. Complete the a new table of the same form as above, leaving out the unknown variable. d Hence state the interest rate required to make the annuity last 40 years. Give your answer correct to two decimal places. Recurrence relations 8 Lachlan received an inheritance of \$100\,000. He invests the money at 8\% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws \$9000 and the amount remaining in the account is invested for another year. a How much is in the account at the end of the first year? b Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_0. c What is the value of the investment at the end of year 10? d By the end of which year will the annuity have run out? 9 Christa wins a prize of \$80\,000. She invests the money at 12\% per annum with interest compounded monthly at the end of each month. At the start of each month, before interest is earned, Christa withdraws \$1100 and the amount remaining in the account is invested. a How much interest is earned in the first month? b How much is in the account at the end of the second month? c Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n months, and an initial condition A_0. d What is the value of the investment at the end of the 8th month? e By the end of which month will the annuity have run out? 10 Gwen received an inheritance of \$150\,000. She invests the money at 6\% per annum with interest compounded annually at the end of the year. After the interest is paid, Gwen withdraws \$10\,000 and the amount remaining in the account is invested for another year. a How much is in the account at the end of the first year? b Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_0. c By the end of which year will the annuity have run out? d What amount should be withdrawn at the end of each year so that the balance remains at \$150\,000?

e

If Gwen was only able to invest the money at 4\% per annum, but still withdrew \$10\,000 each year, by the end of which year will the annuity have run out? 11 Tara received an inheritance of \$100\,000. She invests the money at 6\% per annum with interest compounded annually at the end of the year. After the interest is paid, Tara withdraws \$8000 and the amount remaining in the account is invested for another year. a How much is in the account at the end of the first year? b Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n years and an initial condition A_0. c At the end of which year will the annuity have run out? d What amount should be withdrawn at the end of each year so that the balance remains at \$100\,000?

e

If Tara instead withdraws \$12\,000 each year, at the end of which year will the annuity have run out? 12 Vanessa invests \$60\,000 at a rate of 0.5\% per month compounded monthly. Each month she withdraws \$500 from her investment after the interest is paid and the balance is reinvested in the account. a Write a recursive rule for A_n in terms of A_{n - 1} that gives the value of the account after n months and an initial condition A_0. b By the end of which month will the annuity have run out? c If the interest rate was higher and the withdrawals were the same, would the annuity have ended sooner or later? d If the interest rate remained the same and the withdrawals were larger, would the annuity have ended sooner or later? 13 Emma sells her business and with the profit of \$150\,000 sets up an annuity. She will pay herself \$3500 monthly from her annuity which earns interest of 7.6\% per annum compounded quarterly. a Write a recursive rule that gives the value of her annuity, A_n, at the end of quarter n. b Use the sequence facility on your calculator to determine the balance of her annuity after 3 years, correct to the nearest cent. c Use the sequence facility from the previous part to determine after how many years and quarters the annuity will close. d If interest was calculated monthly rather than quarterly, what effect, if any, would this have on the life of her annuity? 14 Eileen invests an inheritance in an annuity from which she makes monthly withdrawals. This annuity pays and calculates her interest annually. The balance of her annuity (in dollars) at the end of the n\text{th} year can be defined recursively as: V_n = \left(1 + 0.064\right) \times V_{n - 1} - 30\,000, V_0 = 280\,000. a How much did Eileen inherit? b What annual interest rate does she earn on her annuity? c What is the monthly amount she withdraws? d What is the balance of her annuity after four years? e During which year will her annuity close? 15 Bill has won \$260\,000 and sets up an annuity earning 4.8\% interest per annum, compounded annually. At the end of each year Bill withdraws \$18\,000. a Complete the table below, using the rounded answer to calculate the amounts for the following year. b If Bill had chosen to withdraw \$18\,000 in twelve monthly instalments of \\$1500 instead of in one lump sum, what would be the result compared to the original situation?

### Outcomes

#### ACMGM099

use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity

#### 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