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.

Worked examples

Finding the present value of an annuity

Straightforward

Problem

Find the present value of receiving $1500 at the end of each year for 3 years, if money earns 6% per annum.

Finding the loan repayment amount

Moderate

Problem

Hannah takes out a loan of $18,000 to be repaid in equal monthly instalments over 36 months (3 years) at 0.5% per month. Find the monthly repayment and total interest paid.

Retirement drawdown: finding equal annual withdrawals

Challenging

Problem

Leon retires with $250,000 in his superannuation account earning 5% per annum. He wants to withdraw equal amounts at the end of each year for 15 years, leaving nothing at the end. Find the annual withdrawal amount.

Practise

Q1·Straightforward

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.

Q2·Straightforward

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.

Q3·Straightforward

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.

Q4·Straightforward

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.

Q5·Moderate

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.

Q6·Moderate

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.

Q7·Moderate

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.

Q8·Moderate

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.

Q9·Moderate

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.

Q10·Challenging

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.

Explanation

Applying

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

Year	Balance
1	$80\,000 \times 1.06 - 12\,000 = \$72\,800.00$
2	$72\,800 \times 1.06 - 12\,000 = \$65\,168.00$
3	$65\,168 \times 1.06 - 12\,000 = \$57\,078.08$
4	$57\,078.08 \times 1.06 - 12\,000 = \$48\,502.77$
5	$48\,502.77 \times 1.06 - 12\,000 = \$39\,412.93$
6	$39\,412.93 \times 1.06 - 12\,000 = \$29\,777.71$
7	$29\,777.71 \times 1.06 - 12\,000 = \$19\,564.37$
8	$19\,564.37 \times 1.06 - 12\,000 = \$8\,738.23$

After 8 withdrawals, $8,738.23 remains in the fund.

Q11·Challenging

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

Q12·Challenging

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

Open Math

Present value of annuities

Financial Mathematics · MS-F5

Name:

Date:

Q1Straightforward

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.

Q2Straightforward

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.

Q3Straightforward

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.

Q4Straightforward

Q5Moderate

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.

Q6Moderate

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.

Q7Moderate

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.

Q8Moderate

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.

Q9Moderate

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.

Q10Challenging

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.

Q11Challenging

(1.04)^5 = 1.2166529

and

(1.04)^{10} = 1.4802443

Q12Challenging

(1.04)^{20} = 2.19112

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