Victoria invests \$400\,000 in an annuity paying 3.2\% interest per annum. The annuity is designed to give her an annual payment of \$47\,372 for 10 years.
The amortisation table for this annuity is shown below in dollars:
| \text{Payment} \\ \text{number } (n) | \text{Start} \\ \text{Balance } | \text{Interest} \\ \text{earned} | \text{Payment} \\ \text{made} | \text{Reduction}\\ \text{in principal} | \text{Balance of} \\ \text{annuity} |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 400\,000.00 |
| 1 | 400\,000 | 12\,800.00 | 47\,372.00 | 34\,572.00 | |
| 2 | 11\,693.70 | 47\,372.00 | 35\,678.30 | 329\,749.70 | |
| 3 | 329\,749.70 | 10\,551.99 | 47\,372.00 | 36\,820.11 | 292\,929.69 |
| 4 | 292\,929.69 | 9373.75 | 47\,372.00 | 37\,998.25 | 254\,931.44 |
| 5 | 254\,931.44 | 8157.81 | 47\,372.00 | 215\,717.25 | |
| 6 | 215\,717.25 | 6902.95 | 47\,372.00 | 40\,469.05 | 175\,248.20 |
| 7 | 175\,248.20 | 5607.94 | 47\,372.00 | 41\,764.06 | 133\,484.14 |
| 8 | 133\,484.14 | 47\,372.00 | 90\,383.63 | ||
| 9 | 90\,383.63 | 2892.28 | 47\,372.00 | 44\,479.72 | 45\,903.91 |
| 10 | 45\,903.91 | 1468.93 | 47\,372.00 | 45\,903.07 | 0.84 |
Find the balance of the annuity after one payment has been made.
Find the reduction in the principal of the annuity after payment number 5.
Find the amount of interest earned in the 8th year.
A university student is given a living allowance of \$6000 for her first year of study. She invests the money in an annuity paying an interest rate of 3\% per annum, compounded monthly. From this annuity, she receives a monthly payment of \$508.
The amortisation table for this annuity is given below in dollars:
| \text{Payment} \\ \text{number } (n) | \text{Start} \\ \text{Balance} | \text{Interest} \\ \text{earned} | \text{Payment} \\ \text{made} | \text{Reduction}\\ \text{in principal} | \text{End} \\ \text{Balance} |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 6000.00 |
| 1 | 6000.00 | 15.00 | 508.00 | 493.00 | 5507.00 |
| 2 | 5507.00 | 13.77 | 508.00 | 494.23 | 5012.77 |
| 3 | 5012.77 | 12.53 | 508.00 | 495.47 | 4517.30 |
| 4 | 4517.30 | 11.29 | 508.00 | 496.71 | 4020.59 |
| 5 | 4020.59 | 10.05 | 508.00 | 497.95 | 3522.64 |
| 6 | 3522.64 | A | 508.00 | B | C |
| 7 | 7.56 | 508.00 | 500.44 | 2523.01 | |
| 8 | 2523.01 | 6.31 | 508.00 | 501.69 | 2021.32 |
| 9 | 2021.32 | 5.05 | 508.00 | 502.95 | 1518.37 |
| 10 | 1518.37 | 3.80 | 508.00 | 504.20 | 1014.17 |
| 11 | 1014.17 | 2.54 | 508.00 | 505.46 | 508.71 |
| 12 | 508.71 | 1.27 | 508.00 | 506.73 | 1.98 |
Find the monthly interest rate as a percentage.
Find the interest when Payment 1 is received.
Find the reduction in principal when Payment 3 is received.
Find the balance of the annuity after Payment 5 has been received.
Find the value of:
A
B
C
Find the value of the last payment if the balance of the annuity is to be zero after all payments have been received.
Find the total return from the annuity.
Find the total amount of interest earned.
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.
Find the value of the following:
\text{N}
\text{I}\%
\text{PV}
\text{Pmt}
\text{FV}
\text{P/Y}
\text{C/Y}
At the end of which year will the annuity have run out?
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.
Find the value of the following:
\text{N}
\text{I}\%
\text{PV}
\text{Pmt}
\text{FV}
\text{P/Y}
\text{C/Y}
At the end of which month will the annuity have run out?
Katrina has \$150\,000 to invest. She wishes to withdraw \$1400 each month after the interest is paid for a period of 30 years. Assume that the interest is compounded monthly.
Find the value of the following:
\text{N}
\text{I}\%
\text{PV}
\text{Pmt}
\text{FV}
\text{P/Y}
\text{C/Y}
Hence, determine the amount of the annual interest rate Katrina needs if she wants her investment to last 30 years. Round your answer to two decimal places.
Avril invests \$190\,000 at a rate of 7\% per annum compounded annually. Determine what Avril's annual withdrawal should be if she wants the investment to last 25 years.
Victoria invests \$190\,000 at a rate of 12\% per annum compounded monthly. Determine what Victoria's equal monthly withdrawal should be if she wants the investment to last 20 years.
Hannah invests \$190\,000 at a rate of 16\% per annum compounded quarterly. Determine what Hannah's quarterly withdrawal should be if she wants the investment to last 30 years.
Homer received an inheritance of \$200\,000. He invests the money at 7\% per annum with interest compounded annually at the end of the year. After the interest is paid, Homer withdraws \$17\,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 + 1} in terms of A_n that gives the value of the account after (n + 1) years, and an initial condition A_0.
What is the value of the investment at the end of year 10?
Use the graphing facility on your calculator to graph the balance of the loan, A, against the year n.
At the the end of which year will the annuity run out?
At the end of which year will the balance of the loan be \$170\,000?
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+1} in terms of A_{n} that gives the value of the account after n months and an initial condition A_0.
At the the end of which month will the annuity run out?
Use the graphing facility on your calculator to graph the balance of the loan, A, against the month n.
Use the graph to estimate how long it will take for the balance of the loan be \$35\,000.
Caitlin invests \$90\,000 at a rate of 1.5\% per month compounded monthly. Each month she withdraws \$1650 from her investment after the interest is paid and the balance is reinvested in the account.
Write a recursive rule for A_{n + 1} in terms of A_n that gives the value of the account after (n + 1) months, and an initial condition A_0.
At the the end of which month will the annuity 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?
Use the graphing facility on your calculator to graph the balance of the loan, A, against the month n.
Graph the balance of the loan, A, against the month n, if the interest rate was increased to 1.7\%.
Was your prediction in part (c) correct? Explain your answer.
Graph the balance of the loan, A, against the month n, if the withdrawals were increased to \$1800 with the original interest rate.
Was your prediction in part (c) correct? Explain your answer.
Judy received an inheritance of \$500\,000. She invests the money at 4.5\% per annum with interest compounded annually at the end of the year. After the interest is paid, Judy withdraws \$25\,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 + 1} in terms of A_n that gives the value of the account after (n + 1) years, and an initial condition A_0.
Use the graphing facility on your calculator to graph the balance of the loan, A, against the year n.
At the the end of which year will the annuity run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$500\,000?
If Judy was only able to invest the money at 3.5\% per annum, but still withdrew \$25\,000 each year, by the end of which year will the annuity run out?
Tina received an inheritance of \$200\,000. She invests the money at 6\% per annum with interest compounded annually at the end of the year. After the interest is paid, Tina withdraws \$14\,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 + 1} in terms of A_n that gives the value of the account after (n + 1) years, and an initial condition A_0.
At the the end of which year will the annuity run out?
What amount should be withdrawn at the end of each year so that the balance remains at \$200\,000?
If Tina instead withdraws \$18\,000 each year, by the end of which year will the annuity have run out?
Rosethe wins a prize of \$50\,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 Rosethe withdraws \$700 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 + 1} in terms of A_n that gives the value of the account after (n + 1) months, and an initial condition A_0.
What is the value of the investment at the end of the 8th month?
At the the end of which month will the annuity have run out?
Noah has been granted a small scholarship of \$12\,500 to attend a sporting institute for one year. The money is invested in an annuity that pays 8\% p.a., compounded quarterly. Noah is paid \$3243 per quarter from the annuity.
Write a recurrence relation to describe this situation.
Determine how much money is left in the annuity at the end of the year.
How much should Noah be paid in his last payment so that he uses all of his remaining balance before the end of the year?
Complete the amortisation table below for Noah's annuity:
| \text{Payment} \\ \text{number } (n) | \text{Payment} | \text{Interest} | \text{Reduction}\\ \text{in principal} | \text{Balance of} \\ \text{annuity} |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 12\,500 |
| 1 | 3243.00 | 2993.00 | 9507.00 | |
| 2 | 3243.00 | 190.14 | 6454.14 | |
| 3 | 3243.00 | 129.06 | 3113.92 | |
| 4 | 66.80 | 3340.22 | 0 |