Describe what an annuity is.
Determine whether the following are types of annuities:
An account in which you make regular contributions and the interest is paid at the end of each period.
An account in which you make contributions when you have spare money and the interest is paid at the beginning of each period.
An account from which you withdraw contributions that decrease as the balance decreases.
An account in which you make regular withdrawals and the interest is paid at the end of each period.
Is the future value of an annuity greater than or equal to the sum of the individual contributions?
Determine whether the following options describe an annuity:
In order to buy a car, Eileen opens an investment account with a bank and initially deposits \$5000, with plans to make regular contributions of \$800 at the end of each month.
David takes out a home loan of \$1\,000\,000 and plans to pay it off using monthly instalments that decrease over the term of the loan.
Yvonne saves her weekly pocket money in a piggy bank in order to buy a computer.
Charlie invests \$100\,000 and withdraws \$1000 from it each month as it accrues interest.
Joanne deposited \$800\,000 into a new investment account. After 2 months she withdrew \$2500, and every 2 months after that she increased her withdrawal amount by \$50.
Is Joanne's investment an annuity? Explain your answer.
A deposit of \$4000 is made on June 1, 2006 into an investment account and a deposit of \$400 is made each year on May 31.
The balance at the end of each 12-month period for this investment, where interest is compounded annually, is given by:
\text{New balance} = 1.06 \times \text{Old balance} + 400, and \text{Initial balance} = 4000.
State the annual interest rate.
Determine the balance on June 1, 2007.
Determine the value of the investment on June 1, 2008.
At the start of each month, Oliver deposits \$2000 into a savings account to help him set money aside for bills. This savings account earns 12\% p.a. interest, compounded monthly.
The table below shows the first few months of 2013. All values in the table are in dollars.
Complete the row for June.
| Month | Balance at the beginning of month | Monthly payment | Interest | Balance at the end of month |
|---|---|---|---|---|
| \text{March} | 30\,000 | 2000 | 320 | 32\,320 |
| \text{April} | 32\,320 | 2000 | 343.20 | 34\,663.20 |
| \text{May} | 34\,663.20 | 2000 | 366.63 | 37\,029.83 |
| \text{June} |
At the start of each month, Oliver deposits \$3000 into a savings account to help him set money aside for bills. This savings account earns 24\% p.a. interest, compounded monthly.
The table below shows the first few months of 2018. All values in the table are in dollars.
Complete the row for June.
| Month | Balance at the beginning of month | Monthly payment | Interest | Balance at the end of month |
|---|---|---|---|---|
| \text{March} | 40\,000 | 3000 | 860 | 43\,860 |
| \text{April} | 43\,860 | 3000 | 937.20 | 47\,797.20 |
| \text{May} | 47\,797.20 | 3000 | 1015.94 | 51\,813.14 |
| \text{June} |
To save for a deposit on a house, Yuri sets aside \$2000 at the start of each month into a savings account that earns 12\% p.a. interest, compounded monthly.
The table below shows a few months of 2016. All values in the table are in dollars.
Complete the row for October.
| Month | Balance at the beginning of month | Monthly payment | Interest | Balance at the end of month |
|---|---|---|---|---|
| \text{August} | 30\,000 | 2000 | 320 | 32\,320 |
| \text{September} | 32\,320 | 2000 | 343.20 | 34\,663.20 |
| \text{October} | ||||
| \text{November} | 37\,029.83 | 2000 | 390.30 | 39\,420.13 |
To save for a deposit on a house, Neville started to set aside \$2000 at the start of each month into a savings account that earns 12\% p.a. interest, compounded monthly.
The table below shows a few months of 2017. All values in the table are in dollars.
Complete the row for October.
| Month | Balance at the beginning of month | Monthly payment | Interest | Balance at the end of month |
|---|---|---|---|---|
| \text{August} | 40\,000 | 1000 | 410 | 41\,410 |
| \text{September} | 41\,410 | 1000 | 424.10 | 42\,834.10 |
| \text{October} | ||||
| \text{November} | 44\,272.44 | 1000 | 452.72 | 45\,725.16 |
Nadia initially deposits \$7000 into an investment account. At the end of each quarter, Nadia makes an extra deposit of \$500.
The table below shows the first few quarters of 2015. All values in the table are in dollars.
| Month | Balance at the beginning of quarter | Interest | Deposit | Balance at the end of quarter |
|---|---|---|---|---|
| \text{Jan-Mar} | 7000 | 140 | 500 | 7640 |
| \text{Apr-Jun} | 7640 | 152.80 | 500 | 8292.80 |
| \text{Jul-Sep} | 8292.80 | 165.86 | 500 | 8958.66 |
| \text{Oct-Dec} |
Use the numbers for the January quarter to calculate the quarterly interest rate.
Calculate the annual interest rate of her investment.
Complete the row for the October quarter.
Victoria initially deposits \$9000 into a savings bank account. At the end of each quarter, Victoria makes an extra deposit of \$600.
The table below shows the first few quarters of 2011. All values in the table are in dollars.
| Month | Balance at the beginning of quarter | Interest | Deposit | Balance at the end of quarter |
|---|---|---|---|---|
| \text{Jan-Mar} | 9000 | 360 | 600 | 9960 |
| \text{Apr-Jun} | 9960 | 398.40 | 600 | 10\,958.40 |
| \text{Jul-Sep} | 10\,958.40 | 438.34 | 600 | 11\,996.74 |
| \text{Oct-Dec} |
Use the numbers for the January quarter to calculate the quarterly interest rate.
Calculate the annual interest rate of her investment.
Complete the row for the October quarter.
Sharon initially deposits \$7000 into an investment account. At the end of each quarter, she makes an extra deposit of \$600.
The table below shows the first three quarters of the investment. All values in the table are in dollars.
| Month | Balance at the beginning of quarter | Interest | Deposit | Balance at the end of quarter |
|---|---|---|---|---|
| \text{Jan-Mar} | 7000 | 210 | 600 | 7810 |
| \text{Apr-Jun} | 7810 | 234.30 | 600 | 8644.30 |
| \text{Jul-Sep} | 8644.30 | 259.33 | 600 | 9503.63 |
| \text{Oct-Dec} |
Use the numbers for January quarter to calculate the quarterly interest rate.
Calculate the annual interest rate of her investment.
Complete the row for the October quarter.
The table below shows the first few years of an investment with regular deposits:
| Year | Beginning Balance | Interest | Deposit | End Balance |
|---|---|---|---|---|
| 1 | y | 180 | 600 | 9780 |
| 2 | 9780 | 195.60 | 600 | 10\,575.60 |
| 3 | 10\,575.60 | x | 600 | 11\,387.11 |
| 4 | 11\,387.11 | 227.74 | 600 | w |
Find the value of:
x
y
Using the numbers for Year 1, calculate the annual interest rate.
The table below shows the first few years of an investment with regular deposits:
| Year | Beginning balance | Interest | Deposit | Deposit |
|---|---|---|---|---|
| 1 | y | 100 | 600 | 5700 |
| 2 | 5700 | 114.00 | 600 | 6414.00 |
| 3 | 6414.00 | x | 600 | 7142.28 |
| 4 | 7142.28 | 142.85 | 600 | w |
Find the value of:
w
x
y
Using the numbers for Year 1, calculate the annual interest rate.
David opens an investment account with a starting balance of \$4000. The account awards interest of 6\% p.a., compounded at the end of each month. He initially plans to make regular contributions of \$300 at the start of each month.
| Month | Balance at beginning of month | Deposit at start of month | Interest compounded at end of month | Balance at end of month |
|---|---|---|---|---|
| \text{Jan} | 4000 | 300 | 21.50 | 4321.50 |
| \text{Feb} |
Complete the table for the month of February.
David decided to change his investment approach and deposit half as much twice as often, as shown in the table below.
| Month | Balance at beginning of month | Deposit at start of month | Deposit at half way through the month | Interest compounded at end of month | Balance at end of month |
|---|---|---|---|---|---|
| \text{Jan} | 4000 | 150 | 150 | ||
| \text{Feb} | 150 | 150 |
Complete the table for this new approach.
Does the new approach increase, decrease, or have no effect on the final value of the investment?
The spreadsheet below shows the first year of an annuity with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Year} | \text{Beginning Balance} | \text{Interest} | \text{Deposit} | \text{End Balance} |
| 2 | 1 | 7000 | 910 | 600 | 8510 |
| 3 | |||||
| 4 | |||||
| 5 |
Calculate the annual interest rate for this annuity.
Write a formula for cell \text{B5}.
Write a formula for cell \text{C6}.
Write a formula for cell \text{E4}.
The spreadsheet below shows the first year of an annuity with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Year} | \text{Beginning balance} | \text{Interest} | \text{Deposit} | \text{End balance} |
| 2 | 1 | 9000 | 810 | 500 | 10\,310 |
| 3 | |||||
| 4 | |||||
| 5 |
Calculate the annual interest rate for this investment.
Write a formula for cell \text{B6}.
Write a formula for cell \text{C3}.
Write a formula for cell \text{E5}.
Consider the start of an annuity model contained in the spreadsheet below:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | \text{Starting Balance} | \text{Interest Earned} | \text{Deposit} | \text{Closing Balance} |
| 2 | 5000 | 50 | 100 | 5150 |
| 3 |
Which column will not increase in value as we move down the spreadsheet?
What will be the value in the cell \text{A3}?
If we continue the model, which cell will have the same value as cell \text{A10}?
Will the interest earned increase, decrease or stay the same as we move down the spreadsheet?
The spreadsheet below shows the first month of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Initial Investment} | 30\,000 | |||
| 2 | \text{Annual Interest Rate} | 0.108 | |||
| 3 | \text{Monthly Deposit} | 600 | |||
| 4 | |||||
| 5 | |||||
| 6 | \text{Month} | \text{Beginning Balance} | \text{Interest} | \text{Deposit} | \text{End Balance} |
| 7 | 1 | 30\,000 | 270 | 600 | 30\,870 |
| 8 | |||||
| 9 | |||||
| 10 |
Calculate the monthly interest rate for this investment.
Write a formula for cell \text{B7} in terms of one or more cells.
Write a formula for cell \text{C7}.
Write a formula for cell \text{E7}.
The spreadsheet below shows the first month of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Initial Investment} | 40\,000 | |||
| 2 | \text{Annual Interest Rate} | 0.072 | |||
| 3 | \text{Monthly Deposit} | 500 | |||
| 4 | |||||
| 5 | |||||
| 6 | \text{Month} | \text{Beginning balance} | \text{Interest} | \text{Deposit} | \text{End balance} |
| 7 | 1 | 40\,000 | 240 | 500 | 40\,740 |
| 8 | |||||
| 9 | |||||
| 10 |
Calculate the monthly interest rate for this investment.
Write the formula for cell \text{B7} in terms of one or more cells.
Write the formula for cell \text{C7}.
Write the formula for cell \text{E7}.