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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.    \(r=\dfrac{4 \%}{2}=2 \%\ \text{per 6 months}\)

\(\text {Compounding periods}=3 \times 2=6\)

\(\Rightarrow \text {Factor }=6.4343\)

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

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

  1. `$34\ 140`
  2. `$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.

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

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

Show Answers Only
  1. `$45\ 097.17`
  2. `$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 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 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.

  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
  1. `$3589.09`
  2. `$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 33%.

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, 2ADV M1 EQ-Bank 1

Ralph opens an annuity account and makes a contribution of $12 000 at the end of each year for 9 years.

For the first 8 years, the interest rate is 4% per annum, compounded annually.

For the 9th year, the interest rate decreases to 3% per annum, compounded annually.
 


 

Use the Future Value of an Annuity table to calculate the amount in the account immediately after the 9th contribution is made.  (3 marks)

Show Answers Only

`$125\ 885.04`

Show Worked Solution

`text(After 8 years:)`

`text(Total)` `= 9.214 xx $12\ 000`
  `= $110\ 568`

 
`text{After 9 years (9th contribution made year end):}`

`text(Total)` `= 110\ 568 xx 1.03 + 12\ 000`
  `= $125\ 885.04`

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)

Show Answers Only

`$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 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.

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

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

Show Answers Only
  1. `$64\ 995.60`
  2. `$4995.60`
Show Worked Solution

i.   `text(FV factor = 5.4163)`

`:.\ text(FV 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)` `=\ text(FV − total repayments)`
    `= 64\ 995.60 – (5 xx 12\ 000)`
    `= $4995.60`

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.
 

2UG-2005-26b
 

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

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

Show Answers Only
  1. `$15\ 516.36`
  2. `$1116.36`
Show Worked Solution

i.   `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`

 

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

Financial Maths, STD2 F5 SM-Bank 4

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)
 

 

Show Answers Only

`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 2014 HSC 21 MC

A table of future value interest factors is shown.

2014 21 mc

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? 

  1.    `$50\ 500`
  2.    `$51\ 000`
  3.    `$103\ 040`
  4.    `$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{# 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)
  2UG-2011-27d1

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. 
 

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  $407100  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
  1. `$32\ 342`
  2. `$50\ 000`
  3. `$285.70`
Show Worked Solution

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

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


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

♦ Mean mark 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`

 

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

♦♦ Mean mark 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)`