SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

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)

Show Answers Only

\($306.78\)

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.
 

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

  3. 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.
  4. 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)

Show Answers Only

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 2023 HSC 25

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

  1. Micky wants to save $450 000 over the next 10 years.
  2. 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)

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

Show Answers Only

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 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.

  1. 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)

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

Show Answers Only

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)

Show Answers Only

`$ 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 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. 
 

2ug-2016-hsc-q28_31
 

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)

Show Answers Only

`$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)`.

2015 30c

  1. 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)

  2. 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)

Show Answers Only

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 2009 HSC 27a

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

2UG-2009-27a

  1. 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)

  2. 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)

  3. 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)

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)`