At the start of each month, Michael deposits \$3000 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:
| Month | Monthly Deposit | Balance at beginning of month | Interest | Balance at end of month |
|---|---|---|---|---|
| \text{March} | 3000 | 30\,000 | 330 | 33\,330 |
| \text{April} | 3000 | 33\,000 | 363.30 | 36\,693.30 |
| \text{May} | 3000 | 36\,693.30 | 396.93 | 40\,090.23 |
| \text{June} |
Complete the row for June.
Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the account after n + 1 months, and an initial condition V_0.
Use the sequences facility on calculator to determine the balance at the beginning of August 2014, assuming no withdrawals have been made.
Tara initially deposits \$7000 into an investment account. At the end of each quarter, Tara makes an extra deposit of \$600.
The table below shows the first few quarters of 2014. All values in the table are in dollars.
| Month | Balance at beginning of quarter | Interest | Deposit | Balance at 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 the January quarter to calculate the quarterly interest rate.
Calculate the nominal annual interest rate of her investment.
Complete the row for the October quarter.
Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the account after n + 1 quarters, and an initial condition V_0.
Determine total value of her savings at the beginning of 2016.
The table below shows the first few years of an investment with regular deposits. All values are in dollars.
| Year | Beginning balance | Interest | Deposit | End balance |
|---|---|---|---|---|
| 1 | y | 320 | 700 | 9020 |
| 2 | 9020 | 360.80 | 700 | 10\,080.80 |
| 3 | 10\,080.80 | x | 700 | 11\,184.03 |
| 4 | 11\,184.03 | 447.36 | 700 | w |
Find the value of:
w
x
y
Using the numbers for Year 1, calculate the annual interest rate.
Write a recursive rule for V_n in terms of V_{n - 1} that gives the value of the account after n years and an initial condition V_0.
The total amount of this investment is withdrawn after 25 years. Determine the value of the investment when it is withdrawn.
Calculate the total interest earned over the 25 years.
The table below shows the first few months of an investment with regular deposits. All values are in dollars.
| Year | Beginning balance | Interest | Deposit | End balance |
|---|---|---|---|---|
| \text{Jan 2015} | y | 100 | 200 | 40\,300 |
| \text{Feb 2015} | 40\,300 | 100.75 | 200 | 40\,600.75 |
| \text{Mar 2015} | 40\,600.75 | x | 200 | 40\,902.52 |
| \text{Apr 2015} | 40\,902.25 | 102.26 | 200 | 41\,204.51 |
Calculate the monthly interest rate.
Find the value of x.
Find the value of y, the initial investment.
Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the account after n + 1 months, and an initial condition V_0.
The total amount of this investment is withdrawn after 6 years. Determine the value of the investment when it is withdrawn.
Calculate the total interest earned over the 6 years.
Consider the two investment options below:
Deposit a lump-sum of \$45\,000 at a rate of 7.5\% p.a. compounded annually for 25 years.
Deposit \$1800 annually in an annuity, at a rate of 7.5\% p.a. compounded annually for 25 years.
After 25 years, how much will the lump-sum investment be worth?
After 25 years, how much will the annuity be worth?
After 25 years, how much more will you have from the lump-sum investment than from the annuity?
Consider the two investment options below:
Deposit a lump-sum of \$14\,300 at a rate of 6\% p.a. compounded annually for 25 years.
Deposit \$1300 annually in an annuity, at a rate of 6\% p.a. compounded annually for 25 years.
After 25 years, how much will the lump-sum investment be worth?
After 25 years, how much will the annuity be worth?
After 25 years, how much more will you have from the annuity investment than from the lump-sum investment?
When James was born, his parents set up a trust fund to help him pay for his university education. James’s parents opened an account earning 5.3\% per annum compounded annually. They deposit \$300 at the end of each year, with the first amount deposited on James's first birthday.
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 in James’s account after his 18th birthday.
To save up to buy a car, Roxanne opens a savings account that earns 7\% per annum compounded monthly. She initially deposits \$1600 when she opens the account at the beginning of the month, and then deposits \$125 at the end of every month.
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 Roxanne has saved after 2 years.
To save up for a garden renovation, Luke and Vanessa open a savings account. They intend to have enough money to renovate their garden in 4 years time. The account earns 6\% p.a. compounded quarterly, and they need to save \$26\,000.
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 Luke and Vanessa should deposit each quarter.
Aaron is saving for a deposit to buy a house. He can initially afford to deposit \$11\,000, thanks to a donation from his parents, and will then make deposits of \$1200 at the end of each month. He wants to choose a savings account that is compounded monthly in order to save \$90\,000 in 5 years time.
Find the value of the following:
\text{N}
\text{I}\%
\text{PV}
\text{Pmt}
\text{FV}
\text{P/Y}
\text{C/Y}
Hence, state the annual interest rate Aaron should search for. Give your answer to two decimal places.
Susana needs to save \$74\,000 to start a small business. A bank offers a savings account for this purpose, offering 14\% p.a. compounded monthly. Susana will initially deposit \$8000 and can afford to contribute an extra \$1100 to the account at the end of every month.
Find the value of the following:
\text{N}
\text{I}\%
\text{PV}
\text{Pmt}
\text{FV}
\text{P/Y}
\text{C/Y}
Hence, determine at the end of which month Susana will achieve her goal.
A deposit of \$3000 is made on June 1, 2007 into an investment account and a deposit of \$200 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 A_{n + 1} = 1.06 A_n + 200, and A_0 = 3000.
State the annual interest rate.
Determine the balance on June 1, 2008.
Determine the value of the investment on June 1, 2014.
Mr. McCoy opened a bank account for his granddaughter Lisa on the day she was born: January 5, 2005. He deposited \$4000 into the account. Mrs. McCoy, Lisa’s grandmother, also deposited money into this account on that day, and continues to do so by depositing \$300 every 3 months.
The balance at the end of each quarter for this investment, where interest is compounded quarterly, is given by A_{n + 1} = 1.04 A_n + 300, A_0 = 4300.
State the quarterly interest rate.
Determine the nominal annual interest rate.
Determine the balance on the day after Lisa's first birthday.
Determine the balance on the day after Lisa's 12th birthday.
Luigi opens an account to help save for a house. He opens the account at the beginning of 2013 with an initial deposit of \$30\,000 that is compounded annually at a rate of 4.2\% per annum. He makes further deposits of \$1000 at the end of each year.
How much money is in the account at the end of the first year?
Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the account after n + 1 years, and an initial condition V_0.
Find the total value of his savings at the beginning of 2019.
Hence, find how much interest Luigi has earned.
Judy opens a savings account to motivate herself to save regularly. She opens the account on September 2, 2012 with the intention of making regular deposits of \$190 on the 2nd of each month. The interest rate for this account is 12\% per annum with monthly compounding.
Find the monthly interest rate of this account.
If she first invests \$3000 when she opens the account and makes no further deposits in September 2012, find the balance at the end of the first month.
Find the value of her savings account at the end of the second month.
Write a recursive rule for V_{n + 1} in terms of V_n that gives the value of the account after n + 1 months, and an initial condition V_0.
Find the total value of her savings account on September 1, 2018.
Row 2 of the following spreadsheet models the first year of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Year} | \text{Beginning balance} | \text{Interest} | \text{Deposit} | \text{Ending balance} |
| 2 | 1 | 6000 | 660 | 500 | 7160 |
| 3 | 2 | ||||
| 4 | 3 | ||||
| 5 | 4 |
Calculate the annual interest rate for this investment.
Write a formula for cell \text{B3}.
Write a formula for cell \text{C6} in terms of \text{B6}.
Write a formula for cell \text{E5}.
Using a spreadsheet application, reproduce this spreadsheet and determine the end balance for the 4th year.
Calculate the total interest earned over the 4 years.
Row 7 of the following spreadsheet models the first month of an investment with regular deposits:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | \text{Initial Investment} | 20\,000 | |||
| 2 | \text{Annual Interest Rate} | 0.072 | |||
| 3 | \text{Monthly Deposit} | 400 | |||
| 4 | |||||
| 5 | |||||
| 6 | \text{Month} | \text{Beginning Balance} | \text{Interest} | \text{Deposit} | \text{End Balance} |
| 7 | 1 | 20\,000 | 120 | 400 | 20\,520 |
| 8 | 2 | ||||
| 9 | 3 | ||||
| 10 | 4 |
Calculate the monthly interest rate for this investment.
Write a formula for cell \text{B7}.
Write a formula for cell \text{C7}.
Write a formula for cell \text{E7}.
Using a spreadsheet application, reproduce this spreadsheet and determine the end balance for the 5th year.
Calculate the total interest earned over the 60 months.