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Financial Maths, STD2 2020 HSC 37 (Adapted)

Estelle deposits a single lump sum into an account that earns 3% per annum compound interest.

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

  

From this account, Estelle plans to make the following withdrawals.

  • $2000 at the end of each year for the first 15 years (the first withdrawal is one year after the deposit).
  • $5000 at the end of each year for a further 10 years, that is, in years 16 to 25.

Find the smallest lump sum Estelle must deposit so that both sets of withdrawals can be made.   (3 marks)

Show Answers Only

\(\text{Lump sum required}=\$51\,251\)

Show Worked Solution

\(\text{Annuity 1:}\ \ PV\ \text{of}\ \ \$2000\ \text{annuity for 15 years at}\ r=0.03\)

\(\Rightarrow PV\ \text{factor}=11.938\)

\(\therefore\ PV\ \text{Annuity 1}=11.938\times 2000=\$23\,876\)
  

\(\text{Annuity 2:}\ \ PV\ \text{of}\ \ \$5000\ \text{annuity for years 16−25 at}\ r=0.03\)

\(PV\ \text{Annuity 2}\) \(=PV(25\ \text{years})-PV(15\ \text{years})\)  
  \(=5000\times 17.413-5000\times 11.938\)  
  \(=5000\times(17.413-11.938)\)  
  \(=\$27\,375\)  

 
\(\therefore\ \text{Lump sum required}=23\,876+27\,375=\$51\,251\)

Financial Maths, STD2 2025 HSC 34 (Adapted)

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

Larry invests a single amount of $18 000 for 5 years at 9% per annum, compounding monthly.

Tobias wants to end up with the same amount as Larry by using an annuity. He will pay a fixed sum into an account at the end of each month for 5 years, with the account also paying 9% per annum, compounding monthly.

Using the table, work out how much Tobias must deposit each month.   (3 marks)

Show Answers Only

\(\$373.65\)

Show Worked Solution

\(r=\dfrac{0.09}{12}=0.0075,\ \ n=12\times 5=60\)

\(\text{Larry’s investment:}\)

\(FV=18\,000(1+0.0075)^{60}=28\,182.26\)
  

\(\text{Tobias’s investment:}\)

\(\text{Annuity factor:}\ 75.42414\)

\(\text{Annuity}\times 75.42414\) \(=\$28\,182.26\)
\(\text{Annuity}\) \(=\dfrac{28\,182.26}{75.42414}=\$373.65\)

Financial Maths, STD2 2021 HSC 31 (Adapted)

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 Paula $600 000 to buy an apartment, with interest charged at 1.8% per annum compounding monthly. She agrees to repay the loan in equal monthly repayments over a 25-year period.

What monthly repayment is needed to repay the loan in 25 years? Give your answer correct to the nearest cent.   (2 marks)

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\(\text{Monthly repayment}=\$2485.11\)

Show Worked Solution

\(\text{Monthly interest rate}\ (r)=\dfrac{1.8}{12}\%=0.15\%=0.0015\)

\(N=25\times 12=300\)

\(\Rightarrow\ \text{PV annuity factor}=241.43789\)
 

\(\therefore\ \text{Monthly repayment}=\dfrac{600\,000}{241.43789}=\$2485.11\)

Financial Maths, STD2 2023 HSC 25 (Adapted)

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

  
 

  1. Sue wants to save $180 000 over the next 5 years. The account pays 8% per annum, compounding annually.
  2. Using the table, find the amount Sue should contribute each year, to the nearest dollar.   (2 marks)
  3. Instead, Sue decides to contribute $7500 every three months for 5 years into an account paying 8% per annum, compounding quarterly.
  4. Using the table, find how much Sue will have at the end of 5 years.   (3 marks)
Show Answers Only

a.    \(\$30\,680\)

b.    \(\$182\,227.50\)

Show Worked Solution

a.    \(\text{Applicable interest rate}=8\%\)

\(\text{Compounding periods}=5\times 1=5\)

\(\Rightarrow\ \text{Factor}=5.867\)

\(\therefore\ \text{Contribution (annual)}=\dfrac{180\,000}{5.867}=\$30\,680\ \text{(nearest dollar)}\)
 

b.    \(\text{Applicable interest rate}=\dfrac{8\%}{4}=2\%\ \text{per quarter}\)

\(\text{Compounding periods}=5\times 4=20\)

\(\Rightarrow\ \text{Factor}=24.297\)

\(\therefore\ \text{Total (after 5 years)}=7500\times 24.297=\$182\,227.50\)

Financial Maths, STD2 2021 HSC 21 (Adapted)

Trevor opens a savings account with $8000. The account pays interest at a fixed monthly rate. At the end of each month the interest is added, and Trevor then deposits a further $400.

The spreadsheet below records the first six months of the account, together with the start of the seventh month.

 

By first finding the monthly interest rate, complete the row for month 7.   (3 marks)

Show Answers Only

\(\text{Monthly rate}=0.2\%\)

\(\text{Month 7: beginning}=\$10\,508.52,\ \text{interest}=\$21.02,\ \text{end}=\$10\,929.54\)

Show Worked Solution

\(\text{Monthly interest rate}=\dfrac{16.00}{8000}=0.002=0.2\%\)

\(\text{Row 7 calculations:}\)

\(\text{Beginning balance}=\$10\,508.52\)

\(\text{Monthly interest}=10\,508.52\times 0.002=\$21.02\)

\(\text{End of month balance}\) \(=10\,508.52+21.02+400\)
  \(=\$10\,929.54\)

Financial Maths, STD2 2024 HSC 20 (Adapted)

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

  
 

  1. Callum invests $300 at the end of each year for 6 years into an account earning 4% per annum, compounded annually. Using the table, calculate the future value of Callum’s investment.   (1 mark)
  2. Devi wants to have saved $6200 in 4 years. She will make equal payments at the end of every six months into an account paying 6% per annum, compounded six-monthly.
  3. Using the table, find the minimum amount Devi must pay each six months. Give your answer to the nearest $10 and support it with calculations.   (2 marks)
Show Answers Only

a.    \(\$1989.90\)

b.    \($700\ \text{(nearest \$10)}\)

Show Worked Solution

a.    \(\text{6 annual periods at 4% p.a.}\Rightarrow\text{Factor}=6.6330\)

\(FV=300\times 6.6330=\$1989.90\)
  

b.    \(r=\dfrac{6\%}{2}=3\%\ \text{per 6 months}\)

\(\text{Compounding periods}=4\times 2=8\)

\(\Rightarrow\ \text{Factor}=8.8923\)
 

\(\text{Find minimum payment:}\)

\(6200\) \(=\text{Payment}\times 8.8923\)
\(\text{Payment}\) \(=\dfrac{6200}{8.8923}=697.23\ldots=$700\ \text{(nearest \$10)}\)

Financial Maths, STD2 EO-Bank 28

Leon opens a superannuation account to build up savings for retirement. At the end of each year he pays in $4000, and the account earns 5% per annum, compounded annually.

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

  
 

  1. Write down the formula used in cell C9, using appropriate grid references.   (1 mark)

  2. Determine the value that belongs in cell C9.   (1 mark)
  3. Starting from the end of year 4, Leon lifts his yearly payment from $4000 to $7000. Find the balance in the account at the end of year 7, and state how much larger this is than if he had stayed with $4000 payments.   (3 marks)
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a.    \(\text{=E8*B3}\)

b.    \(\text{C9}=\$410.00\)

c.    \(\text{Balance at end of year 7}=\$42\,025.54\)

\(\text{Leon has}\ \$9457.50\ \text{more than at the standard contribution.}\)

Show Worked Solution

a.    \(\text{Formula: =E8*B3}\)
 

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

\(\text{C9}=8200\times 0.05=\$410.00\)
  

c.    \(\text{Using}\ \ P+I+C\ \ \text{from end of year 4 balance}\ \$17\,240.50:\)

\(\text{Increased contributions of}\ \$7000\ \text{from year 5:}\)

\(\text{Year 5:}\ 17\,240.50+17\,240.50\times 0.05+7000=\$25\,102.53\)

\(\text{Year 6:}\ 25\,102.53+25\,102.53\times 0.05+7000=\$33\,357.66\)

\(\text{Year 7:}\ 33\,357.66+33\,357.66\times 0.05+7000=\$42\,025.54\)
  

\(\text{Standard contributions of}\ \$4000\ \text{from year 5:}\)

\(\text{Year 5:}\ 17\,240.50+17\,240.50\times 0.05+4000=\$22\,102.53\)

\(\text{Year 6:}\ 22\,102.53+22\,102.53\times 0.05+4000=\$27\,207.66\)

\(\text{Year 7:}\ 27\,207.66+27\,207.66\times 0.05+4000=\$32\,568.04\)
  

\(\text{Difference}=42\,025.54-32\,568.04=\$9457.50\)

\(\therefore\ \text{Leon has}\ \$9457.50\ \text{more by increasing his contributions.}\)

Financial Maths, STD2 EO-Bank 19

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

  
 

Rina and Owen each set up an annuity, depositing a fixed amount at the end of every year.

  1. Rina pays $2500 each year for 5 years into an annuity earning 3% per annum, compounded annually. Using the table, find the present value of Rina’s annuity.   (1 mark)
  2. Owen pays $4000 each year for 3 years into an annuity earning 5% per annum, compounded annually. Whose annuity has the greater present value? Justify your answer with calculations.   (2 marks)
Show Answers Only

a.    \(\$11\,449.25\)

b.    \(\text{Rina’s annuity is greater: }\$11\,449.25>\$10\,892.80\)

Show Worked Solution

a.    \(\text{Table factor when}\ n=5,\ r=3\%\ \Rightarrow\ 4.5797\)

\(\therefore\ PVA\ \text{(Rina)}=2500\times 4.5797=\$11\,449.25\)
  

b.    \(\text{Table factor when}\ n=3,\ r=5\%\ \Rightarrow\ 2.7232\)

\( PVA\ \text{(Owen)}=4000\times 2.7232=\$10\,892.80\)

\(\text{Rina’s annuity is greater: }\$11\,449.25>\$10\,892.80\)

Financial Maths, STD2 2025 HSC 18 (Adapted)

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

A scholarship fund receives a contribution of $4000 at the end of each year for 15 years. The fund earns 5% per annum, compounded annually.

Using the table, calculate the value of the fund at the end of 15 years.   (2 marks)

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\(\$86\,316\)

Show Worked Solution

\(r=5\%\ \text{annually}\)

\(\text{Compounding periods}=15\)

\(\text{Annuity factor}=21.579\)

\(\therefore\ FV\ \text{(annuity)}=4000\times 21.579=\$86\,316\)

Financial Maths, STD2 2018 HSC 26c (Adapted)

Sofia contributes $175 to an annuity at the end of every month and plans to keep this up for 3 years.

Ignoring interest, how much will Sofia have paid into the annuity in total over the 3 years?   (1 mark)

Show Answers Only

\(\text{Total contributed}=\$6300\)

Show Worked Solution

\(\text{Total contributed}=3\times 12\times 175=\$6300\)

Financial Maths, STD2 2013 23 MC (Adapted)

Elias opens a savings account that pays 4% per annum, compounded quarterly. At the end of every quarter he deposits $800, beginning one quarter after the account is opened, and continues for the following 18 months.

How much is in Elias’s account at the end of the 18 months?

  1. $4800.00
  2. $4921.61
  3. $4970.83
  4. $5095.30
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Interest: 4% p.a.}\ \ \Rightarrow\ \  \text{1% per quarter}\)

\(\text{18 months}=6\ \text{end-of-quarter deposits}\)

\(\text{Value of 1st deposit}=800(1.01)^5=840.81\)

\(\text{Value of 2nd deposit}=800(1.01)^4=832.48\)

\(\text{Value of 3rd deposit}=800(1.01)^3=824.24\)

\(\text{Value of 4th deposit}=800(1.01)^2=816.08\)

\(\text{Value of 5th deposit}=800(1.01)^1=808.00\)

\(\text{Value of 6th deposit}=800\)
   

\(\therefore\ \text{Amount in account}\)

\(=840.81+832.48+824.24+816.08+808.00+800.00\)

\(=\$4921.61\)

\(\Rightarrow B\)