Annuities

Calculate the future and present value of annuities using recurrence relations and tables; apply to superannuation accumulation, loan repayments and retirement drawdowns.

Financial Mathematics · MS-F5

Sub-topics

Future value of annuities

Recognise an annuity as a series of equal periodic contributions; calculate future value using a recurrence relation or the formula

FV = M \cdot \frac{(1+r)^n - 1}{r}

; apply to superannuation and savings goals.

Present value of annuities

Calculate the present value of an annuity using the formula

PV = M \times \frac{1-(1+r)^{-n}}{r}

; relate present and future value; find regular payment amounts; model loan repayments and retirement drawdowns.

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Annuities

Financial Mathematics · MS-F5

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Future value of annuities

Q1Straightforward

Petra deposits $500 at the end of each year into a savings account earning 4% per annum. Use the recurrence relation

A_{n+1} = A_n \times 1.04 + 500

with

A_0 = 0

to find the balance at the end of year 3.

Q2Straightforward

Use the future value formula

FV = M \times \dfrac{(1+r)^n - 1}{r}

to find the future value of an annuity with monthly contributions of $1000, an annual interest rate of 5% (compounded annually), and 3 annual contributions. Give your answer to the nearest cent.

Q3Straightforward

$400 is deposited at the end of each half-year into an account earning 4% per annum, compounded half-yearly (2% per half-year). Use the future value formula with

r = 0.02

and

n = 4

to find the account balance after 2 years.

Q4Moderate

Sam deposits $200 at the end of each month into a savings account earning 6% per annum, compounded monthly (0.5% per month). Find the total accumulated after 2 years (24 months), to the nearest cent.

Q5Moderate

$1000 is invested at the end of each quarter into a fund earning 4% per annum, compounded quarterly (1% per quarter). Find the total accumulated after 3 years (12 quarters), to the nearest cent.

Q6Moderate

An employee contributes $2000 to their superannuation account at the end of each year. The fund earns 5% per annum, compounded annually. Find the accumulated balance after 5 years, to the nearest cent.

Q7Moderate

A saver deposits $400 at the end of each 6-month period for 3 years. The account earns 4% per annum, compounded half-yearly (2% per half-year). The future value interest factor for

r = 2\%

and

n = 6

periods is

6.3081

. Use this factor to find the future value of the annuity, to the nearest cent.

Q8Moderate

Alicia deposits $300 at the end of each month for 5 years into an account earning 6% per annum, compounded monthly (0.5% per month). Calculate the total interest earned over this period, to the nearest cent.

Q9Challenging

Jason deposits $600 at the end of each month into an account earning 6% per annum, compounded monthly (0.5% per month). After how many full months will his balance first exceed $3000? Use the recurrence

A_{n+1} = A_n \times 1.005 + 600

with

A_0 = 0

Q10Challenging

Priya wants to accumulate $100\,000 in her superannuation account by making equal monthly deposits at the end of each month. The fund earns 6% per annum, compounded monthly (0.5% per month). Using

FV = M \times \dfrac{(1+r)^n - 1}{r}

with

n = 120

months and

(1.005)^{120} = 1.8194

, find the required monthly deposit, to the nearest cent.

Q11Challenging

Anna has two options for a 15-year investment at 5% per annum:

**Option A:** Invest a lump sum of $15\,000 today (compound interest:

A = P(1+r)^n

).

**Option B:** Contribute $1000 at the end of each year for 15 years (annuity formula).

Given

(1.05)^{15} = 2.07893

, calculate the difference in the final values (to the nearest cent). Which option gives more, and by how much?

Q12Challenging

Ben contributes $600 at the end of each month to his superannuation account. The fund earns 6% per annum, compounded monthly (0.5% per month). After how many full months will his balance first exceed $10\,000? Use the recurrence

A_{n+1} = A_n \times 1.005 + 600

with

A_0 = 0

Present value of annuities

Q13Straightforward

Use the present value formula

PV = M \times \dfrac{1-(1+r)^{-n}}{r}

to find the present value of 3 annual payments of $1000 at 5% per annum. Give your answer to the nearest cent.

Q14Straightforward

A government bond will pay $15,000 in 6 years. If the current interest rate is 5% per annum, what is the present value of this payment today? Give your answer to the nearest cent.

Q15Straightforward

A savings plan pays $800 at the end of each year for 4 years. Using the present value interest factor of

3.4651

for

r = 6\%

and

n = 4

, find the present value of the plan.

Q16Straightforward

Maria has $80,000 in a retirement account earning 4% per annum. She wants to make equal annual withdrawals over 5 years, leaving nothing in the account at the end. Find the annual withdrawal amount, to the nearest cent.

Q17Moderate

Jamie borrows $20,000 to buy a car. The loan charges 1% interest per month and is repaid in equal monthly instalments over 24 months. Find the monthly repayment, to the nearest cent.

Q18Moderate

A loan of $15,000 is to be repaid in equal quarterly instalments over 2 years (8 quarters) at 2% interest per quarter. Find the quarterly repayment amount, to the nearest cent.

Q19Moderate

Using your answer from the previous question (quarterly repayment of $2047.65 on a $15,000 loan at 2% per quarter for 8 quarters), calculate the total interest paid over the life of the loan.

Q20Moderate

Calculate the present value of receiving $10,000 at the end of each year for 5 years, if money can earn 4% per annum. Give your answer to the nearest cent.

Q21Moderate

A car dealer advertises a car at $800 per month for 48 months at 0.5% per month interest. The present value interest factor for

r = 0.5\%

and

n = 48

42.5803

. Find the equivalent cash price of the car.

Q22Challenging

Paul has $80,000 in a retirement fund earning 6% per annum. He withdraws $12,000 at the end of each year. Use the recurrence

A_{n+1} = A_n \times 1.06 - 12\,000

with

A_0 = 80\,000

to find the balance remaining after 8 withdrawals.

Q23Challenging

Sandra invests a lump sum of $200,000 at 4% per annum for 5 years. At the end of 5 years, the accumulated amount is used to fund equal annual withdrawals over the following 10 years (also at 4% per annum). Find the annual withdrawal amount, to the nearest cent.

Given:

(1.04)^5 = 1.2166529

and

(1.04)^{10} = 1.4802443

Q24Challenging

A lottery prize can be paid as $20,000 per year for 20 years (payments at end of each year), or as a lump sum today. If money earns 4% per annum, calculate by how much the present value of the annual payment option exceeds a lump sum offer of $250,000. Give your answer to the nearest dollar.

Given:

(1.04)^{20} = 2.19112

Worked solutions and answers at openmath.au/year-12/standard-2/annuities