Annuities and superannuation
Calculate future and present values of annuities using geometric series; model superannuation accumulation; solve annuity problems using recurrence relations.
Worked examples
Future value of an annuity using geometric series
Straightforward
Problem
Deposits of $2000 are made at the end of each year for 4 years into an account earning 5% per annum. Find the accumulated value at the end of 4 years.
1
Write the future value of each deposit, noting each earns interest for a different number of years (the 1st deposit grows for 3 years, the 2nd for 2 years, the 3rd for 1 year, and the 4th for 0 years).
Future values: 1st deposit , 2nd deposit , 3rd deposit , 4th deposit .
2
Recognise this as a geometric series and sum it.
This is a geometric series with first term , common ratio , and terms.
Using the annuity formula directly:
Answer
The accumulated value after 4 years is .
Superannuation accumulation
Moderate
Problem
Yuki contributes $5000 at the end of each year into a superannuation fund earning 6% per annum for 20 years. Find the total value of the fund at retirement.
1
Identify the annuity parameters.
, , .
2
Apply the future value formula.
3
Evaluate.
Answer
The fund is worth approximately at retirement.
Present value and retirement drawdowns
Challenging
Problem
On retirement, Yuki transfers her $183,928 into a new account earning 4% per annum and makes equal annual withdrawals for 15 years until the account is exhausted. Find the annual withdrawal.
1
Set up the present value equation: the present value of the withdrawal annuity must equal the fund balance.
2
Evaluate the annuity factor.
3
Solve for .
Answer
Yuki can withdraw approximately per year for 15 years.
Practise
Q1·Straightforward
An annuity account satisfies the recurrence relation , with . Find , the value of the account after 3 contributions.
Explanation
Q2·Straightforward
Use the future-value formula to find the future value of an annuity with annual contribution , interest rate per annum, over years. Give your answer in dollars, rounded to the nearest cent.
Explanation
Q3·Straightforward
Alicia deposits $500 at the end of each year for 3 years into an account earning 4\% per annum. How much is in the account at the end of 3 years? Give your answer in dollars, rounded to the nearest cent.
Explanation
Q4·Straightforward
What is the present value of an annuity that pays $1500 per year for 3 years, if the interest rate is 4\% per annum? Use . Round to the nearest cent.
Explanation
Q5·Moderate
Raj contributes $3000 at the end of each year into a superannuation fund earning 6\% per annum. Find the total value of the fund after 10 years. Round to the nearest cent.
Explanation
(to 5 decimal places)
Q6·Moderate
Deposits of $1000 are made at the end of each of 4 years into an account paying 5\% per annum. Explain why the accumulated value is a geometric series, and find the total. Round to the nearest cent.
Explanation
At the end of year 4, the accumulated value of each deposit is:
- Year 1 deposit:
- Year 2 deposit:
- Year 3 deposit:
- Year 4 deposit:
This is a geometric series with first term , common ratio , and terms:
- Year 1 deposit:
- Year 2 deposit:
- Year 3 deposit:
- Year 4 deposit:
This is a geometric series with first term , common ratio , and terms:
Q7·Moderate
How much must be deposited at the end of each year for 5 years, into an account earning 4\% per annum, to accumulate $20{,}000? Round your answer to the nearest cent.
Explanation
Q8·Moderate
How much must be in a superannuation fund today to allow annual withdrawals of $20{,}000 for 10 years, starting in one year, if the fund earns 5\% per annum? Round to the nearest cent.
Explanation
Q9·Challenging
Mei has $5000 in a savings account and also deposits an additional $1000 at the end of each year. The account earns 5\% per annum. How much will she have after 5 years? Round to the nearest cent.
Explanation
**Lump sum (future value):**
**Annuity (future value):**
**Total:**
**Annuity (future value):**
**Total:**
Q10·Challenging
Priya deposits $4000 at the end of each year into an account earning 6\% per annum. After how many full years will the fund first exceed $50{,}000?
Explanation
We need:
Check :
Check : ✓
The fund first exceeds $50{,}000 after full years.
Check :
Check : ✓
The fund first exceeds $50{,}000 after full years.
Q11·Challenging
Leo invests $2000 per year for 5 years into an account earning 4\% per annum. After 5 years he stops making deposits, but the account continues to earn interest at 6\% per annum for a further 5 years. Find the total value of the account after 10 years. Round to the nearest cent.
Explanation
**Phase 1 — accumulation (years 1–5 at 4\%):**
**Phase 2 — growth (years 6–10 at 6\%, no further contributions):**
**Phase 2 — growth (years 6–10 at 6\%, no further contributions):**
Q12·Challenging
Sam contributes $6000 at the end of each year into a superannuation fund earning 7\% per annum for 30 years. On retirement, the entire fund is transferred to an account earning 4\% per annum. Sam then makes equal annual withdrawals for 20 years before the fund is exhausted. Find the annual withdrawal Sam can afford. Round to the nearest cent.
Explanation
**Step 1 — accumulation phase ($6000/year, 7\%, 30 years):**
**Step 2 — drawdown phase (4\%, 20 years):**
**Step 2 — drawdown phase (4\%, 20 years):**
Open Math
Annuities and superannuation
Algebra / Functions · MAV-12-08
Name:
Date:
Q1Straightforward
An annuity account satisfies the recurrence relation , with . Find , the value of the account after 3 contributions.
Q2Straightforward
Use the future-value formula to find the future value of an annuity with annual contribution , interest rate per annum, over years. Give your answer in dollars, rounded to the nearest cent.
Q3Straightforward
Alicia deposits $500 at the end of each year for 3 years into an account earning 4\% per annum. How much is in the account at the end of 3 years? Give your answer in dollars, rounded to the nearest cent.
Q4Straightforward
What is the present value of an annuity that pays $1500 per year for 3 years, if the interest rate is 4\% per annum? Use . Round to the nearest cent.
Q5Moderate
Raj contributes $3000 at the end of each year into a superannuation fund earning 6\% per annum. Find the total value of the fund after 10 years. Round to the nearest cent.
Q6Moderate
Deposits of $1000 are made at the end of each of 4 years into an account paying 5\% per annum. Explain why the accumulated value is a geometric series, and find the total. Round to the nearest cent.
Q7Moderate
How much must be deposited at the end of each year for 5 years, into an account earning 4\% per annum, to accumulate $20{,}000? Round your answer to the nearest cent.
Q8Moderate
How much must be in a superannuation fund today to allow annual withdrawals of $20{,}000 for 10 years, starting in one year, if the fund earns 5\% per annum? Round to the nearest cent.
Q9Challenging
Mei has $5000 in a savings account and also deposits an additional $1000 at the end of each year. The account earns 5\% per annum. How much will she have after 5 years? Round to the nearest cent.
Q10Challenging
Priya deposits $4000 at the end of each year into an account earning 6\% per annum. After how many full years will the fund first exceed $50{,}000?
Q11Challenging
Leo invests $2000 per year for 5 years into an account earning 4\% per annum. After 5 years he stops making deposits, but the account continues to earn interest at 6\% per annum for a further 5 years. Find the total value of the account after 10 years. Round to the nearest cent.
Q12Challenging
Sam contributes $6000 at the end of each year into a superannuation fund earning 7\% per annum for 30 years. On retirement, the entire fund is transferred to an account earning 4\% per annum. Sam then makes equal annual withdrawals for 20 years before the fund is exhausted. Find the annual withdrawal Sam can afford. Round to the nearest cent.
Worked solutions and answers at openmath.au/year-12/advanced/financial-mathematics/annuities-and-superannuation