Annuities (Y12)

Financial Maths, STD2 EQ-Bank 29

Mei sets up a superannuation account to save for retirement. She contributes $5000 at the end of each year into the account which earns interest at 6% per annum, compounded annually.

The spreadsheet shown models the first 4 years of the account.

Calculate the value in cell C9. (1 mark)
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From the end of year 4, Mei increases her annual contribution from $5000 to $8000. Calculate the balance in her superannuation account at the end of year 7, and determine how much more this is than if she had continued contributing $5000. (3 marks)
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a. $\text{C9} = \$618.00$

b. $\text{Balance at end of year 7} = \$51\,519.99$

$\text{Mei has}\ \$9550.80\ \text{more than at the standard contribution.}$

Show Worked Solution

a. $\text{C9 (Year 3 interest)} = \text{balance at start} \times \text{rate}$

$\text{C9} = 10\,300 \times 0.06 = \$618.00$

b. $\text{Using}\ \ P+I+C\ \ \text{from end of year 4 balance}\ \$21\,873.08:$

$\text{Increased contributions of}\ \$8000\ \text{from year 5:}$

$\text{Year 5:}\ 21\,873.08+21\,873.08 \times 0.06+8000=\$31\,185.46$

$\text{Year 6:}\ 31\,185.46+31\,185.46 \times 0.06+8000=\$41\,056.59$

$\text{Year 7:}\ 41\,056.59+41\,056.59 \times 0.06+8000=\$51\,519.99$

$\text{Standard contributions of}\ \$5000\ \text{from year 5:}$

$\text{Year 5:}\ 21\,873.08+21\,873.08 \times 0.06+5000=\$28\,185.46$

$\text{Year 6:}\ 28\,185.46+28\,185.46 \times 0.06+5000=\$34\,876.59$

$\text{Year 7:}\ 34\,876.59+34\,876.59 \times 0.06+5000=\$41\,969.19$

$\text{Difference}=51\,519.99-41\,969.19=\$9550.80$

$\therefore\ \text{Mei has}\ \$9550.80\ \text{more by increasing her contributions.}$

Financial Maths, STD2 EQ-Bank 27

Haruki opens a savings account. He deposits $2000 at the end of each year into an account earning 3% per annum, compounded annually.

Complete the table below to show the growth of Haruki's savings over the first 4 years. (3 marks)

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Hence, calculate the total interest Haruki earns over the 4 years. (1 mark)

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b. $\$367.25$

Show Worked Solution

a. $\text{Year 2:}$

$\text{Interest} = 2000 \times 0.03 = \$60.00$

$\text{Closing balance} = 2000+60+2000 = \$4060.00$

$\text{Year 3:}$

$\text{Interest} = 4060 \times 0.03 = \$121.80$

$\text{Closing balance} = 4060+121.80+2000 = \$6181.80$

$\text{Year 4:}$

$\text{Interest} = 6181.80 \times 0.03 = 185.454\ldots = \$185.45$

$\text{Closing balance} = 6181.80+185.45+2000 = \$8367.25$

b. $\text{Total deposits} = 4 \times 2000 = \$8000.00$

$\text{Total interest} = 8367.25-8000 = \$367.25$

Financial Maths, STD2 F5 2025 HSC 18

A table of future value interest factors for an annuity of $1 is shown.

The prize in a lottery is an annuity of $5000 a year for 10 years, invested at 4.5% per annum compounding annually.

What will be the value of the prize at the end of 10 years? (2 marks)

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$$61\,440$

Show Worked Solution

$ r=4.5\%\ \text{annually}$

$\text{Compounding periods = 10}$

$\text{Annuity factor = 12.288}$

$\therefore\ \text{FV (annuity)}\ = 5000 \times 12.288=$61\,440$

Financial Maths, STD2 F5 2025 HSC 34

The table shows future value interest factors for an annuity of $1.

Lin invests a lump sum of $21 000 for 7 years at an interest rate of 6% per annum, compounding monthly.

Yemi wants to achieve the same future value as Lin by using an annuity. Yemi plans to deposit a fixed amount into an investment account at the end of each month for 7 years. The investment account pays 6% per annum, compounding monthly.

Using the table provided, determine how much Yemi needs to deposit each month. (3 marks)

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$$306.78$

Show Worked Solution

$r=\dfrac{0.06}{12}=0.005, \ n=12 \times 7=84$

$\text{Lin’s investment:}$

$F V=21\,000(1+0.005)^{84}=31\,927.76$

♦ Mean mark 43%.

$\text{Yemi’s investment:}$

$\text{Annuity factor:} \ 104.07393$

$\text{Annuity} \times 104.07393$	$=$31\,927.76$
$\text{Annuity}$	$=\dfrac{31\,927.76}{104.07393}=$306.78$

Financial Maths, STD2 F5 2024 HSC 20

The table shows the future value for an annuity of $1 for varying interest rates and time periods.

Ken invests $200 at the start of each year for eight years, at an interest rate of 5% per annum.
Calculate the future value of Ken's investment. (1 mark)

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Shay is planning to take a holiday in three years. She needs $4500 for this holiday and will make regular six-monthly payments into an account that earns interest at the rate of 4% per annum, compounded 6 monthly.
What is the minimum amount Shay needs to pay into this account every 6 months? Give your answer to the nearest $10. Support your answer with calculations. (2 marks)

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a. $F V=\$ 2005.32$

b. $\$700$

Show Worked Solution

a. $\text {8 annual periods at 5% p.a.} \Rightarrow \text { Factor}=10.0266$

$F V=200 \times 10.0266=\$ 2005.32$

b. $r=\dfrac{4 \%}{2}=2 \%\ \text{per 6 months}$

$\text {Compounding periods}=3 \times 2=6$

$\Rightarrow \text {Factor }=6.4343$

$4500$	$=\ \text{Annuity} \times 6.4343$
$\text{Annuity}$	$=\dfrac{4500}{6.4343}=$699.38$
	$=\$700\ \text{(nearest \$10)}$

♦ Mean mark (b) 51%.

Financial Maths, STD2 F5 2024 HSC 41

Twenty-five years ago, Phoenix deposited a single sum of money into a new bank account, earning 2.4% interest per annum compounding monthly.

Present value interest factors for an annuity of $1 for various interest rates $(r)$ and numbers of periods $(n)$ are given in the table.

Phoenix made the following withdrawals from this account.

$2000 at the end of each month for the first 15 years, starting at the end of the first month.
$1200 at the end of each month for the next 10 years, starting at the end of the 181st month after the account was opened.

Calculate the minimum sum that Phoenix could have deposited in order to make these withdrawals. (4 marks)

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$\text{Minimum deposit}\ = $391\,344.80$

Show Worked Solution

$\text{1st Annuity}$

$\text{Find PVA for \$2000 paid monthly for 1st 15 years:}$

$r= \dfrac{2.4%}{12} = 0.2\% = 0.002$

$\text{Total payments (to Phoenix)}\ = 15 \times 12 = 180$

$\text{PVA factor (from table)}\ = 151.036$

$\text{PVA (1st annuity)}\ = 2000 \times 151.036 = $302\,072$

♦♦ Mean mark 39%.

$\text{2nd Annuity}$

$\text{Find PVA for \$1200 paid monthly from year 16 to 25:}$

$\text{PVA (2nd annuity) = PVA (25 years) }-\text{ PVA (15 years)}$

$r= \dfrac{2.4%}{12} = 0.2\% = 0.002$

$\text{Total payments (25 years)}\ = 25 \times 12 = 300$

$\text{PVA factors (from table): 225.430 (25 years), 151.036 (15 years)}$

$\text{PVA (2nd annuity)}$	$=(1200 \times 225.430)-(1200 \times 151.036)$
	$=$89\,272.80$

$\therefore\ \text{Minimum deposit}\ = 302\,072+89\,272.80 = $391\,344.80$

Financial Maths, STD2 F5 2023 HSC 25

A table of future value interest factors for an annuity of $1 is shown.

Micky wants to save $450 000 over the next 10 years.
If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give your answer to the nearest dollar. (2 marks)

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Instead, Micky decides to contribute $8535 every three months for 10 years to an annuity paying 6% per annum, compounding quarterly.
How much will Micky have at the end of 10 years? (3 marks)

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a. `$34\ 140`

b. `$463\ 177.38`

Show Worked Solution

a. `text{Applicable interest rate}\ =6%`

`text{Compounding periods}\ =10xx1=10`

`=>\ text{Factor}\ = 13.181`

`:.\ text{Contribution (annual)}=(450\ 000)/13.181=$34\ 140`

b. `text{Applicable interest rate}\ =(6%)/4=1.5%\ text{per quarter}`

`text{Compounding periods}\ =10xx4=40`

`=>\ text{Factor}\ = 54.268`

`text{Total (after 10 years)}=8535 xx 54.268=$463\ 177.38`

Mean mark (b) 53%.

Financial Maths, STD2 F5 2022 HSC 30

Eli is choosing between two investment options.

A table of future value interest factors for an annuity of $1 is shown.

What is the value of Eli's investment after 10 years using Option 1 ? (2 marks)

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What is the difference between the future values after 10 years using Option 1 and Option 2? (2 marks)

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a. `$45\ 097.17`

b. `$40.87`

Show Worked Solution

a. `text{Monthly r/i}\ = 1.2/12=0.1text{%}\ \ =>\ \ r= 0.001`

`text{Compounding periods}\ (n)=12xx10=120`

`FV`	`=PV(1+r)^n`
	`=40\ 000(1+0.001)^120=$45\ 097.17`

♦ Mean mark (a) 48%.

b. `text{Quarterly r/i}\ = 2.4/4=0.6text{%}\ \ =>\ \ r= 0.006`

`text{Compounding periods}\ (N) =4xx10=40`

`text{Annuity factor (from table) = 45.05630}`

`FV=1000xx45.05630=45\ 056.30`

`text{Difference}=45\ 097.17-45\ 056.30=$40.87`

♦ Mean mark (b) 43%.

Financial Maths, STD2 F5 2022 HSC 25

The table shows the future value of an annuity of $1.

Zal is saving for a trip and estimates he will need $15 000. He opens an account earning 3% per annum, compounded annually.

How much does Zal need to deposit every year if he wishes to have enough money for the trip in 4 years time? (2 marks)

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How much interest will Zal earn on his investment over the 4 years? Give your answer to the nearest dollar. (2 marks)

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a. `$3589.09`

b. `$660`

Show Worked Solution

a. `text{Using the table:}\ r=3text{%},\ \ n=4`

`text{Annuity factor}\ = 4.184`

`text{Let}\ \ A=\ text{amount invested each year}`

`FV`	`=A xx 4.184`
`15\ 000`	`=A xx 4.184`
`:.A`	`=(15\ 000)/4.184=$3585.09`

b. `text{Total payments}\ = 4 xx 3585.09=$14\ 340.36`

`text{Interest earned}`	`=FV-text{total payments}`
	`=15\ 000-14\ 340.36=659.64`
	`=$660\ \ text{(nearest $)}`

♦♦ Mean mark (b) 33%.

Financial Maths, STD2 F5 2021 HSC 31

Present value interest factors for an annuity of $1 for various interest rates (`r`) and numbers of periods (`N`) are given in the table.

A bank lends Martina $500 000 to purchase a home, with interest charged at 1.5% per annum compounding monthly. She agrees to repay the loan by making equal monthly repayments over a 30-year period.

How much should the monthly payment be in order to pay off the loan in 30 years?

Give your answer correct to the nearest cent. (2 marks)

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`$ 1725.60`

Show Worked Solution

`text{Monthly interest rate}\ (r) = 1.5/12 = 0.125text(%) = 0.00125`

♦ Mean mark 43%.

`N = 30 xx 12 = 360`

`=>\ text(PV annuity factor = 289.75411)`

`:.\ text{Monthly payment}= (500\ 000)/289.75411= $1725.60`

Financial Maths, STD2 F5 2021 HSC 40

A table of future value interest factors for an annuity of $1 is shown.

Simone deposits $1000 into a savings account at the end of each year for 8 years. The interest rate for these 8 years is 0.75% per annum, compounded annually.

After the 8th deposit, Simone stops making deposits but leaves the money in the savings account. The money in her savings account then earns interest at 1.25% per annum, compounded annually, for a further two years.

Find the amount of money in Simone's savings account at the end of ten years. (3 marks)

Show Answers Only

`$8419.81`

Show Worked Solution

`text(In 1st 8 years:)`

♦ Mean mark 35%.

`text(Future value factor = 8.2132)`

`text(Value of annuity)= 8.2132 xx 1000= $8213.20`

`text(After 10 years:)`

`text(Value of investment)= 8213.2 xx (1.0125)^2= $8419.81`

Financial Maths, STD2 F5 2021 HSC 21

Julie invests $12 500 in a savings account. Interest is paid at a fixed monthly rate. At the end of each month, after the monthly interest is added, Julie makes a deposit of $500.

Julie has created a spreadsheet to show the activity in her savings account. The details for the first 6 months are shown.

By finding the monthly rate of interest, complete the final row above for the 7th month. (3 marks)

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`15\ 624.20,\ 23.44,\ 16\ 147.64`

Show Worked Solution

`\text{Monthly interest rate} = \frac{18.75}{12\ 500} = 0.0015 =\ text{0.15%}`

♦ Mean mark 43%.

`\text{Row 7 calculations:}`

`\text{Beginning balance}= 15\ 624.20`

`\text{Monthly interest}= 15\ 624.20 \times 0.0015= 23.44`

`\text{End of month balance}= 15\ 624.20 + 23.44 + 500= 16\ 147.64`

Financial Maths, STD2 F5 2020 HSC 37

Wilma deposited a lump sum into a new bank account which earns 2% per annum compound interest.

Present value interest factors for an annuity of $1 for various interest rates (`r`) and numbers of periods (`N`) are given in the table.

Wilma was able to make the following withdrawals from this account.

$1000 at the end of each year for twenty years (starting one year after the account is opened)
$3000 each year for ten years starting 21 years after the account is opened.

Calculate the minimum lump sum Wilma must have deposited when she opened the new account. (3 marks)

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`$ 34 \ 486`

Show Worked Solution

`text{Annuity 1:}\ PV\ text{of $1000 annuity for 20 years at} \ \ r = 0.02`

♦♦♦ Mean mark 23%.

`PV\ text{factor} = 16.351`

`therefore \ PV\ text{Annuity 1}= 16.351 xx 1000= $16 \ 351`

`text{Annuity 2:}\ PV\ text{of $3000 annuity for years 21–30 at} \ \ r = 0.02`

`PV\ text{Annuity 2}`	`= PVtext{(30 years)}-PVtext{(20 years)}`
	`= 3000 xx 22.396-3000 xx 16.351= $ 18 \ 135`

`:.\ text{Lump sum required}= 16 \ 351 + 18\ 135= $ 34 \ 486`

Financial Maths, STD2 F5 2020 HSC 14 MC

An annuity consists of ten payments, each equal to $1000. Each payment is made on 30 June each year from 2021 through to 2030 inclusive.

The rate of compound interest is 5% per annum.

The present value of the annuity is calculated at 30 June 2020.

The future value of the annuity is calculated at 30 June 2030.

Without performing any calculations, which of the following statements is true?

Present value of the annuity < $10 000 < future value of the annuity
$10 000 < present value of the annuity < future value of the annuity
Future value of the annuity < $10 000 < present value of the annuity
$10 000 < future value of the annuity < present value of the annuity

Show Answers Only

`A`

Show Worked Solution

Mean mark 53%.

`PV\ text{(30 June 2020)} < $10\ 000\ \ text{(each payment discounted to 30-Jun-20 value)}`

`FV\ text{(30 June 2030)} => text{annuity has received}\ \ 10 xx $1000`

`text{payments plus interest}`

`therefore \ FV\ text{(30 June 2030)} \ > \ $10\ 000 `

`=> \ A`

Financial Maths, STD2 F5 2019 HSC 42

The table shows the future values of an annuity of $1 for different interest rates for 4, 5 and 6 years. The contributions are made at the end of each year.

An annuity account is opened and contributions of $2000 are made at the end of each year for 7 years.

For the first 6 years, the interest rate is 4% per annum, compounding annually.

For the 7th year, the interest rate increases to 5% per annum, compounding annually.

Calculate the amount in the account immediately after the 7th contribution is made. (3 marks)

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`$15\ 929.30`

Show Worked Solution

`text{Annuity compounding factor (4% for 6 years)} = 6.633`

♦♦ Mean mark 27%.

`:.\ text(Value after 6 years)= 2000 xx 6.633= $13\ 266.00`

`text(At the end of 7th year:)`

`text(Value)`	`= 13\ 266 xx 1.05 + 2000`
	`= 13\ 929.30 + 2000= $15\ 929.30`

Financial Maths, STD2 F5 2013 23 MC

Zina opened an account to save for a new car. Six months after opening the account, she made first deposit of $1200 and continued depositing $1200 at the end of each six month period. Interest was paid at 3% per annum, compounded half-yearly.

How much was in Zina's account two years after first opening it?

$4909.08
$4982.72
$5018.16
$5094.55

Show Answers Only

`A`

Show Worked Solution

`text(Interest: 3% p.a ⇒ 1.5% per 6 months)`

♦ Mean mark 41%.

`text(After 2 years,)`

`text(Value of 1st deposit) = 1200(1.015)^3 = 1254.81`

`text(Value of 2nd deposit) = 1200(1.015)^2 = 1236.27`

`text(Value of 3rd deposit) = 1200(1.015) = 1218`

`text(Value of 4th deposit) = 1200`

`:.\ text(Amount in account after 2 years)`

`= 1254.81 + 1236.27 + 1218 + 1200`

`=$4909.08`

`=> A`

Financial Maths, STD2 F5 2018 HSC 26c

Ali made monthly deposits of $100 into an annuity for 5 years.

Calculate the total amount Ali deposited into the annuity over this period. (1 mark)

Show Answers Only

`$6000`

Show Worked Solution

`text(Total deposited)= 5 xx 12 xx 100= $6000`

Financial Maths, STD2 F5 2017 HSC 27c

A table of future value interest factors for an annuity of $1 is shown.

An annuity involves contributions of $12 000 per annum for 5 years. The interest rate is 4% per annum, compounded annually.

Calculate the future value of this annuity. (1 mark)

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Calculate the interest earned on this annuity. (1 mark)

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i. `$64\ 995.60`

ii. `$4995.60`

Show Worked Solution

i. `FV text( factor = 5.4163)`

`:.\ FV text( of Annuity)= 12\ 000 xx 5.4163= $64\ 995.60`

♦♦ Mean mark part (ii) 22%.

COMMENT: A very poorly answered question dealing with a core concept in this area.

ii.	`text(Interest earned)`	`=FV-text(total repayments)`
		`= 64\ 995.60-(5 xx 12\ 000)= $4995.60`

Financial Maths, STD2 F5 2016 HSC 28d

The table gives the contribution per period for an annuity with a future value of $1 at different interest rates and different periods of time.

Margaret needs to save $75 000 over 6 years for a deposit on a new apartment. She makes regular quarterly contributions into an investment account which pays interest at 3% pa.

How much will Margaret need to contribute each quarter to reach her savings goal? (2 marks)

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`$2865`

Show Worked Solution

`text(Periods) = 6 xx 4 = 24`

♦ Mean mark 40%.

`text(Interest rate) = 1/4 xx 3 = 0.75text(%)`

`=>\ text(Table factor = 0.0382)`

`(text(i.e. 3.82 cents contributed per)`

`text(quarter = $1 after 6 years))`

`:.\ text(Quarterly contribution)`

`= 75\ 000 xx 0.0382= $2865`

Financial Maths, STD2 F5 2015 HSC 30c

The table gives the present value interest factors for an annuity of $1 per period, for various interest rates `(r)` and numbers of periods `(N)`.

Oscar plans to invest $200 each month for 74 months. His investment will earn interest at the rate of 0.0080 (as a decimal) per month.

Use the information in the table to calculate the present value of this annuity. (1 mark)

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Lucy is using the same table to calculate the loan repayment for her car loan. Her loan is `$21\ 500` and will be repaid in equal monthly repayments over 6 years. The interest rate on her loan is 10.8% per annum.

Calculate the amount of each monthly repayment, correct to the nearest dollar. (2 marks)

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a. `$11\ 136.89\ \ text{(nearest cent)}`

b. `$407\ \ text{(nearest dollar)}`

Show Worked Solution

a. `N = 74,\ r = 0.0080`

♦ Mean mark (a) 48%.

`PVtext{(annuity) table factor}\ = 55.68446`

`:.PV\ text(of annuity)`

`= $200 xx 55.68446= $11\ 136.892`

`= $11\ 136.89\ \ text{(nearest cent)}`

b. `text(Over 6 years)`

♦♦ Mean mark (b) 33%.

`N = 6 xx 12 = 72\ text(months)`

`r = 10.8/12 = text(0.9%) = 0.009`

`PVtext{(annuity) table factor}\ =52.82118`

`text(Let)\ $M =\ text(monthly repayment)`

`text(Loan)\ = PV\ text(of annuity)`

`$21\ 500`	`= M xx 52.82118`
`:.\ M`	`= $407.033…= $407\ \ text{(nearest dollar)}`

Financial Maths, STD2 F5 2005 HSC 26b

Rod is saving for a holiday. He deposits $3600 into an account at the end of every year for four years. The account pays 5% per annum interest, compounding annually.

The table shows future values of an annuity of $1.

Use the table to find the value of Rod’s investment at the end of four years. (2 marks)

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How much interest does Rod earn on his investment over the four years? (2 marks)

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a. `$15\ 516.36`

b. `$1116.36`

Show Worked Solution

a. `text(Using the table),\ r =\ text(5% and)\ n = 4`

`text(Annuity factor = 4.3101)`

`:.\ text(Value of investment)`

`= 3600 xx 4.3101= $15\ 516.36`

b. `text(Interest)`	`= text(Value)-text(Contributions)`
	`= 15\ 516.36-(4 xx 3600)= $1116.36`

Financial Maths, STD2 F5 EQ-Bank 28

Dominique wants to save $15 000 to use as spending money when she travels overseas in 2 years' time.

If she invests $3500 at the end of every 6 months into an account earning 4% p.a., compounded half-yearly, will she have enough?

Use the table below to justify your answer. (2 marks)

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`text(Dominique will not have saved)`

`text(enough to reach)\ $15\ 000.`

Show Worked Solution

`text(Interest rate)\ text{(6 monthly)} = text(4%) -: 2 = text(2%)`

`n = 4\ \ \ text{(6 month periods in 2 years)}`

`=>\ text(FVA factor) = 4.122\ \ \ text{(from Table)}`

`text(FVA)= 3500 xx 4.122= $14\ 427`

`:.\ text(Dominique will not have saved enough to)`

`text(reach)\ $15\ 000.`

Financial Maths, STD2 F5 EQ-Bank 30

Camilla buys a car for $21 000 and repays it over 4 years through equal monthly instalments.

She pays a 10% deposit and interest is charged at 9% p.a. on the reducing balance loan.

Using the Table of present value interest factors below, where `r` represents the monthly interest and `N` represents the number of repayments

Calculate the monthly repayment, `$P`, that Camilla must pay to complete the loan after 4 years (to the nearest $). (3 marks)

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Calculate the total interest paid over the life of the loan. (1 mark)

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a. `text(Camilla must repay $470 per month)`

b. `$3660`

Show Worked Solution

a. `text(Deposit) = 10text(%) xx 21\ 000 = $2100`

`text(Loan Value)= 21\ 000-2100= $18\ 900`

`text(Monthly interest rate) = text(9%)/12 = 0.0075`

`text(# Repayments) = 4 xx 12 = 48`

`=>\ text(PVA Factor) = 40.18478\ \ text{(from Table)}`

`text(Monthly repayment)\ ($P)`	`= (18\ 900)/(40.18478)`
	`= 470.32…= $470\ text{(nearest $)}`

`:.\ text(Camilla must repay $470 per month.)`

b. `text(Total Repayments)= 48 xx 470= $22\ 560`

`:.\ text(Interest paid over loan)= 22\ 560-18\ 900= $3660`

Financial Maths, STD2 F5 EQ-Bank 22

The table below shows the present value of an annuity with a contribution of $1.

Fiona pays $3000 into an annuity at the end of each year for 4 years at 2% p.a., compounded annually. What is the present value of her annuity? (1 mark)

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If John pays $6000 into an annuity at the end of each year for 2 years at 4% p.a., compounded annually, is he better off than Fiona? Use calculations to justify your answer. (2 marks)

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a. `$11\ 423.10`

b. `text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

a. `text(Table factor when)\ \ n = 4,\ r = text(2%) \ => \ 3.8077`

`:.\ PVA\ text{(Fiona)}= 3000 xx 3.8077= $11\ 423.10`

b. `text(Table factor when)\ n = 2, r = text(4%)`

`=> 1.8861`

`:.\ PVA\ text{(John)}= 6000 xx 1.8861= $11\ 316.60`

`:.\ text(Fiona will be better off because her)\ PVA`

`text(is higher.)`

Financial Maths, STD2 F5 2014 HSC 21 MC

A table of future value interest factors is shown.

A certain annuity involves making equal contributions of `$25 000` into an account every 6 months for 2 years at an interest rate of 4% per annum.

Based on the information provided, what is the future value of this annuity?

`$50\ 500`
`$51\ 000`
`$103\ 040`
`$106\ 162`

Show Answers Only

`C`

Show Worked Solution

`text(4 contributions of $25 000 made.)`

♦ Mean mark 43%

`text(Annuity period = 6 months)`

`text{Rate (per annuity period)}=(text(4%))/2=text(2%)`

`text{No. Periods = 4 (4 x 6 months = 2 years)}`

`text(Table value = 4.1216)`

`:.\ text(Annuity Value)=4.1216 xx 25\ 000=$103\ 040`

`=> C`

Financial Maths, STD2 F5 2011 HSC 27d

Josephine invests $50 000 for 15 years, at an interest rate of 6% per annum, compounded annually.

Emma invests $500 at the end of each month for 15 years, at an interest rate of 6% per annum, compounded monthly.

Financial gain is defined as the difference between the final value of an investment and the total contributions.

Who will have the better financial gain after 15 years? Using the Table below* and appropriate formulas, justify your answer with suitable calculations. (4 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Josephine – see Worked Solutions)`

Show Worked Solution

♦ Mean mark 42%
COMMENT: Note that compound interest vs annuity comparisons are commonly tested.

`text(Josephine)`

`text(Investment)`	`= 50\ 000 (1 + 0.06)^15`
	`= 50\ 000 (1.06)^15= $119\ 827.91`

`text(Financial gain)= 119\ 827.91-50\ 000= $69\ 827.91`

`text(Emma)`

`text{Monthly interest rate} = text(6%)-:12=text(0.5%)`

`text{# Monthly Payments}=12 xx 15=180`

`=>\ text{Annuity Factor = 290.8187 (from Table)}`

`text(Investment)= 500 xx 290.8187= $145\ 409.35`

`text(Financial gain)`	`= 145\ 409.35\-text(total contributions)`
	`= 145\ 409.35\-(500 xx 12 xx 15)`
	`= 145\ 409.35\-90\ 000= $55\ 409.35`

`:.\ text(Josephine will have the better financial gain.)`

Financial Maths, STD2 F5 2009 HSC 27a

The table shows the future value of a $1 annuity at different interest rates over different numbers of time periods.

What would be the future value of a `$5000` per year annuity at 3% per annum for 6 years, with interest compounding yearly? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
What is the value of an annuity that would provide a future value of `$407\ 100` after 7 years at 5% per annum compound interest? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
An annuity of $1000 per quarter is invested at 4% per annum, compounded quarterly for 2 years. What will be the amount of interest earned? (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `$32\ 342`

b. `$50\ 000`

c. `$285.70`

Show Worked Solution

a. `text(Table factor when)\ \ n = 6,\ \ \ r =\ 3text(%) \ => \ 6.4684`

`:.\ FV= 5000 xx 6.4684= $32\ 342`

b. `text(Table factor when)\ \ n = 7,\ \ \ r =\ text(5%)`

♦ Mean mark (b) 45%
MARKER’S COMMENT: A common error was to multiply $407 100 by 8.1420 rather than divide.

`=> 8.1420`

`text(Let)\ \ A = text(annuity)`

`FV= A xx 8.1420`

`A= (FV)/8.1420= (407\ 100)/8.1420= $50\ 000`

c. `n=8\ \ \ (text(8 quarters in 2 years) )`

♦♦ Mean mark (c) 31%
MARKER’S COMMENT: When questions asked for the interest paid on annuities, remember to subtract the total principal amounts contributed.

`r = text(4%)/4 =\ text{1% per quarter}`

`:.\ text(Table factor) => 8.2857`

`FV=1000 xx 8.2857=$8285.70`

`text(Interest)`	`= FV (text(annuity) )\-text(Principal)`
	`= 8285.70-(8 xx 1000)= $285.70`

`:.\ text(Interest earned is $285.70)`

\(\text{Annuity} \times 104.07393\)	\(=$31\,927.76\)
\(\text{Annuity}\)	\(=\dfrac{31\,927.76}{104.07393}=$306.78\)

\(4500\)	\(=\ \text{Annuity} \times 6.4343\)
\(\text{Annuity}\)	\(=\dfrac{4500}{6.4343}=$699.38\)
	\(=\$700\ \text{(nearest \$10)}\)

\(\text{PVA (2nd annuity)}\)	\(=(1200 \times 225.430)-(1200 \times 151.036)\)
	\(=$89\,272.80\)