7.05 Annuities
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.
Louis made a single deposit into a savings account, with interest compounding quarterly at 4.5\% p.a. The balance in the account after 8 years was \$25\,000.
Calculate the value of the single payment that Louis made, correct to the nearest dollar.
Maria invests \$4000 at the beginning of each year for 3 years in a savings account that pays 8\% p.a. with interest compounded quarterly.
Find the present value of her first deposit in the beginning of the first year.
Find the present value of her second deposit in the beginning of the first year.
Find the present value of her third deposit in the beginning of the first year.
Find the present value of the annuity in the beginning of the first year.
Dave opens a savings account at the start of year and will have invested \$8000 at the end of each year for 3 years. The account pays 6\% p.a. with interest compounded annually.
When the account is opened, calculate the present value of his first deposit.
Calculate the present value of his second deposit.
Calculate the present value of his third deposit.
Calculate the present value of the annuity.
Joaquina invests \$5000 at the end of each year for 4 years in an investment account that pays 5\% per annum with interest compounded monthly.
What value will her first deposit have grown to at the end of the 4 years?
What value will her second deposit have grown to at the end of the 4 years?
What value will her third deposit have grown to at the end of the 4 years?
Find the future value of the annuity.
Each month, Ellie's employer contributes 10\% of her \$60\,000 salary to her superannuation fund. Ellie's superannuation fund gains 6\% p.a. interest, compounded monthly.
Calculate the value of the monthly super contribution.
After one month Ellie receives her first super contribution. How much will this single contribution be worth after she has worked for 6 months?
How much will the second contribution she receives be worth after working for six months?
The value of the third, fourth, fifth and sixth super contribution after working for 6 months are shown in the table:
If Ellie's superannuation had a balance of \$10\,000 before she started working, find its value after working for 6 months?
| Contribution | Value after six months |
|---|---|
| \text{3rd} | \$507.54 |
| \text{4th} | \$505.01 |
| \text{5th} | \$502.50 |
| \text{6th} | \$500 |
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?
Aoife invested \$2000 into an account with interest compounding monthly and the balance after 5 years is \$2698. Calculate the interest rate per annum, correct to the nearest percentage.
Hayden invested \$5000 into an account with interest compounding quarterly and the balance after 6 years is \$8042.20. Calculate the interest rate per annum, correct to the nearest percentage.
Carlton invested \$10\,000 into an account with interest compounding half yearly and the balance after 9 years is \$28\,543.40. Calculate the interest rate per annum, correct to the nearest percentage.
Bebe invests \$9000 at the beginning of each year for 3 years in a savings account that pays 8\% p.a. with interest compounded monthly.
Find the value her first deposit will grow to at the end of the 3 years.
Find the value her second deposit will grow to.
Find the value her third deposit will grow to.
Find the future value of the annuity.
How much interest will Bebe earn on this annuity?
Reda invests \$6000 at the beginning of each year for 3 years in a savings account that pays 5\% p.a. with interest compounded semi-annually.
Find the value her first deposit will grow to at the end of the 3 years.
Find the value her second deposit will grow to.
Find the value her third deposit will grow to.
Find the future value of the annuity.
How much interest will Reda earn on this annuity?
Buzz invests \$3000 at the end of each year for 3 years in a savings account that pays 7\% p.a. with interest compounded annually.
Calculate the value of his first deposit at the end of the 3 years.
Calculate the final value of his second deposit.
Calculate the future value of the annuity.
Calculate the interest Buzz will earn on this annuity.
Vashti contributes \$4000 to her superannuation account at the beginning of 3 consecutive years. The account pays 4\% p.a. with interest compounded quarterly.
Find the value to which her first contribution will grow at the end of the 3 years.
Find the value to which her second contribution will grow.
Find the value to which her third contribution will grow.
What will the value of her superannuation be in 3 years?
How much interest will Vashti earn on this superannuation annuity?
Alex has a superannuation fund of \$500\,000. He withdraws \$20\,000 at the beginning of each year as an income stream in his retirement. The interest compounds monthly on the remaining balance at 3\% per annum.
Calculate the balance in the account at the end of the first year.
By how much did the balance decrease at the end of the first year?
Calculate the interest he received in the first year.
Peta has a superannuation fund of \$1\,000\,000. She plans to withdraw \$10\,000 at the beginning of each month as an income stream in her retirement. The interest compounds monthly on the remaining balance at 2.5\% per annum.
Calculate the balance in the account at the end of the first month.
By how much did the balance decrease at the end of the first month?
How much interest did she receive in the first month?