Find the minimum initial investment required to sustain each of the following annuities:
Annual payments of \$8000 from an annuity earning 7.5\% p.a. compounded annually, for 6 years.
Monthly payments of \$2500 from an annuity earning 6.9\% p.a. compounded monthly, for 2 years.
Quarterly payments of \$5000 from an annuity earning 5.6\% p.a. compounded quarterly, for 3 years.
Weekly payments of \$350 from an annuity earning 5.2\% p.a. compounded weekly, for 5 years.
Azumi wants to return to study and plans to invest in an annuity to provide a regular income while she completes her master’s degree. The annuity earns 5.1\% p.a. compounded monthly.
If she wants an income of \$2000 per month and plans to complete the degree over three years, how much does she need to invest initially?
Azumi invests \$60\, 000 but decides to complete the degree over four years, what is the maximum monthly payment she can withdraw?
Holly wants to take a year off travel and estimates she will need to withdraw \$1200 per month while travelling. Her current savings account earns 4.2\% p.a. compounded monthly.
What is the minimum amount Holly needs to deposit into her account before leaving to sustain the payments?
Write a recurrence relation for this situation, where A_n is the balance at the end of the \\ nth month and A_0 is the initial investment.
Find the amount in the account after 6 months.
Floyd has invested \$250 \, 000 in an annuity earning 8.4\% p.a. compounded quarterly.
What is the maximum amount Floyd can withdraw each quarter if he wants the funds to last 10 years?
Write a recurrence relation for this situation, where A_n is the balance at the end of the \\ nth quarter and A_0 is the initial investment.
Find the amount in the account after 3 quarters.
Find the amount in the account after 5 years.
Siobhan has invested her superannuation of \$420\, 000 in an annuity earning 6.24\% p.a. compounded monthly.
What is the maximum amount Siobhan can withdraw each month if she wants the funds to last 20 years?
Write a recurrence relation for this situation, where A_n is the balance at the end of the \\ nth month and A_0 is the initial investment.
Find the amount in the account after three years.
Over three years, how much would Siobhan withdraw in total.
Hence, find the interest earned by the account in the first three years.
Alister wins a prize of \$250 \, 000 and invests it in an annuity earning 6.4\% p.a. compounded quarterly.
What is the maximum amount Alister can withdraw each quarter if he wants the funds to last 6 years?
If Alister withdraws the payment found in part (a), find the amount in the account after one year.
Find when the balance will first fall below \$100\, 000.
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.
| Year | Balance at start of year | Interest | Annual withdrawal | Balance at end of year |
|---|---|---|---|---|
| 1 | 260\,000 | 12\,480 | 18\,000 | 254\,480 |
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
State the balance at the start of year 2.
Calculate the interest earned for year 2.
State the annual withdrawal for year 2.
Calculate the balance at the end of year 2.
Complete the table.
Ernesto has invested \$52\, 000 in an annuity earning 6\% p.a. compounded monthly.
What is the maximum amount Ernesto can withdraw each month if he wants the funds to last 4 years?
Ernesto decides to withdraw payments of \$1220 at the end of each month. Complete the following table:
| Month | Balance at beginning of month | Interest | Monthly withdrawal | Balance at end of month |
|---|---|---|---|---|
| 1 | 52\,000 | 260 | 1220 | 51\, 040 |
| 2 | 51\, 040 | 1220 | ||
| 3 | 1220 |
How much interest was earned by the account in the first three months?
Find the amount in the account after 2 years.
Youlanda has invested \$320\, 000 in an annuity earning 6.9\% p.a. compounded annually.
What is the maximum amount Youlanda can withdraw each year if she wants the funds to last 12 years?
Youlanda decides to withdraw payments of \$40\, 000 at the end of each year. Complete the following table:
| Year | Balance at beginning of year | Interest | Annual withdrawal | Balance at end of year |
|---|---|---|---|---|
| 1 | 320\, 000 | 22\, 080 | 40\,000 | 302\,080 |
| 2 | 302\,080 | 40\,000 | ||
| 3 | 40\,000 |
At the end of which year does the balance first fall below 50\% of the initial investment?
Gideon has an annuity that earns 7.2\% p.a. compounded monthly. He withdraws monthly payments of \$4500 and at the end of the first month the balance is \$51\,836
What principal investment did Gideon make?
Find the balance of the account after one year.
How much interest was earned in the first year?
Callum wins a lottery prize and invests it in an annuity earning 5.3\% p.a. compounded annually. Callum withdraws \$65\,000 each year.
If the balance at the end of 2 years is \$664\,897.48, how much did Callum win?
Find the balance of the account after 10 years.
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.
How much is in the account at the end of the first year?
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.
What is the value of the investment at the end of year 10?
By the end of which year will the annuity have run out?
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.
How much interest is earned in the first month?
How much is in the account at the end of the second month?
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.
What is the value of the investment at the end of the 8th month?
By the end of which month will the annuity have run out?
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.
How much is in the account at the end of the first year?
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.
By the end of which year will the annuity have run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$150\,000?
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?
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.
How much is in the account at the end of the first year?
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.
At the end of which year will the annuity have run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$100\,000?
If Tara instead withdraws \$12\,000 each year, at the end of which year will the annuity have run out?
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.
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.
By the end of which month will the annuity have run out?
If the interest rate was higher and the withdrawals were the same, would the annuity have ended sooner or later?
If the interest rate remained the same and the withdrawals were larger, would the annuity have ended sooner or later?
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.
Write a recursive rule that gives the value of her annuity, A_n, at the end of quarter n.
Determine the balance of her annuity after 3 years.
Determine after how many years and quarters the annuity will close.
If interest was calculated monthly rather than quarterly, what effect, if any, would this have on the life of her annuity?
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
During which year will her annuity close?
Sean is saving for retirement and has a goal to retire at 65 with \$600\,000. He invests in an annuity earning 8.1\% p.a. compounded monthly.
To reach his goal Sean must make extra payments to his annuity to for the next 10 years to boost his savings by \$185\,000. What monthly payment does Sean need to make to reach his goal?
At 65 Sean rolls over \$600\, 000 into an income stream account earning 7.6\% p.a. compounded quarterly. If he wants the funds to last until he is 90, what is the maximum quarterly withdrawal he can make?
If Sean withdraws \$13\, 400 from his income stream each quarter, what will the balance of this account be after 10 years?
Fredrick is a 25 year-old who is setting up his first superannuation account. His employer contributes \$1250 to his superannuation fund each quarter. Fredrick’s superannuation fund pays 6.8\% p.a. interest, compounded quarterly.
Assuming the rate and contributions remain constant, find how much has accumulated in his superannuation account after 35 years.
If Frederick aims to retire retires when his superannuation reaches \$1\,400\,000, estimate his age at retirement.
At retirement Frederick invests \$1\,400\,000 into an income stream account earning 5.4\% p.a. compounded monthly. How much can Fredrick withdraw each month if he wants the funds to last 20 years?
Triana is just starting her first job at age 20. Her income is \$56\,000 and her employer pays 9\% of her income per month into a superannuation fund earning 7.5\% p.a. compounded monthly.
What is the monthly payment being made by the employer?
If she retires at 60, estimate the balance of her superannuation from her employer’s contributions.
What assumptions are being made in part (b) and are they realistic?
If Trianna makes her own contributions of \$100 per month into the same account, estimate how much extra will this provide at retirement?
At 60 Triana rolls over \$1\, 500\, 000 into an income stream account earning 5.4\% p.a. compounded monthly. If she wants the funds to last until she is 90, what is the maximum monthly withdrawal she can make?
If Triana withdraws \$12\, 000 per month, at what age will the annuity run out?
Priya wants to return to study and plans to invest in an annuity to provide a regular income while she completes her master’s degree in Physics.
Priya wants an income of \$2600 per month and plans to complete the degree over three years. The account she intends on using has an interest rate of 4.2\% p.a. compounded monthly. Determine how much she needs to invest initially.
If she plans to return to study in 2 years’ time how much will she need to save each week into an account earning 3.12\% p.a. compounded weekly to save for the investment required in part (a)?
Find the difference between her total deposits made into her savings account and the total withdrawals while studying.
What does the difference found in part (c) represent?
Prakash invests \$25\,000 into an annuity and makes additional monthly payments of \$1200 for 8 months. After the 8 months he then makes withdrawals of \$1500 at the end of each month. The bank offers an interest rate of 7.2\% p.a. compounding monthly when no withdrawals are made in a month and 5.7\% p.a. compounded monthly when withdrawals are made.
Construct a recurrence relation to model the first 8 months of the annuity.
Find the balance of the annuity after 8 months.
Construct a recurrence relation to model the balance of the annuity during the withdrawal phase.
Find the balance of the annuity after 5 months of withdrawals.
If the withdrawals continue, after how many withdrawals does the balance fall below the initial investment.
Vivek has decided to set up his own health fund in a savings account, instead of taking out private health insurance. He will deposit fortnightly payments of \$55 into the account, which earns an interest rate of 3.54\% per annum, compounded fortnightly.
In 3 years Vivek finds he needs knee surgery costing \$5800, how far short will his savings fall to pay for the surgery?
If puts the surgery off until he has enough in the account to pay for it, how much longer will he have to wait for the surgery, after the 3 years?
Vivek continues to save and at the age of 58, he has a total of \$75\,800 left in the health fund savings account. Vivek injures his back and needs to take time off work. He decides to invest his savings into an annuity to pay himself an income. The annuity pays 5.73\% interest per annum, compounded monthly. Calculate the monthly income that Vivek can withdraw if he wants the annuity to last for 5 years.
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine which year the annuity will run out.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to find the month during which the annuity will run out.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine the amount of the annual withdrawal.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine the amount of the monthly withdrawal.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine the annual interest rate. Round your answer to two decimal places.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine the amount of the monthly withdrawal.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |
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.
Complete the given table, leaving out the unknown variable.
Use a financial solver to determine how many years the annuity will last. Round your answer to the nearest year.
Fiona would like to work out the interest rate she would need to make the annuity to last 40 years instead.
Complete a new table of the same form as above, leaving out the unknown variable.
Hence, state the interest rate required to make the annuity last 40 years. Round your answer to two decimal places.
| Value | |
|---|---|
| N | |
| I\% | |
| PV | |
| Pmt | |
| FV | |
| P/Y | |
| C/Y |