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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 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)

Show Answers Only

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

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

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

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