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)

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

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

$\Rightarrow PV\ \text{factor}=11.938$

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

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

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