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.

Worked examples

Future value using a recurrence relation

Straightforward

Problem

Kate deposits $800 at the end of each year into a savings account earning 4% per annum. Find the account balance after 3 years using the recurrence

A_{n+1} = A_n \times 1.04 + 800

Future value using the annuity formula

Moderate

Problem

A worker contributes $500 at the end of each month to their superannuation fund for 3 years (36 months). The fund earns 6% per annum, compounded monthly (0.5% per month). Find the accumulated balance at the end of 3 years.

Finding the required monthly contribution

Challenging

Problem

How much must be deposited at the end of each month into an account earning 4.8% per annum (0.4% per month) to accumulate $50\,000 in 10 years (120 months)? Given

(1.004)^{120} = 1.6142

Practise

Q1·Straightforward

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.

Q2·Straightforward

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.

Q3·Straightforward

$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.

Q4·Moderate

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.

Q5·Moderate

$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.

Q6·Moderate

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.

Q7·Moderate

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.

Q8·Moderate

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.

Q9·Challenging

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

Explanation

Applying

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

Month	Balance
1	$600.00
2	$600 \times 1.005 + 600 = \$1203.00$
3	$1203 \times 1.005 + 600 = \$1809.02$
4	$1809.02 \times 1.005 + 600 = \$2417.06$
5	$2417.06 \times 1.005 + 600 = \$3029.14$

After month 4: $2417.06 < $3000

After month 5: $3029.14 > $3000

Jason's balance first exceeds $3000 after **5 months**.

Q10·Challenging

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.

Q11·Challenging

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?

Q12·Challenging

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

Explanation

Applying

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

(key values shown):

Month	Balance
1	$600.00
5	$3030.15
10	$6136.81
14	$8678.53
15	$9321.93
16	$9968.54
17	$10\,618.38

After month 16: $9968.54 < $10\,000

After month 17: $10\,618.38 > $10\,000

Ben's balance first exceeds $10\,000 after **17 months**.

Open Math

Future value of annuities

Financial Mathematics · MS-F5

Name:

Date:

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

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

Q9Challenging

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

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

with

A_0 = 0

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