Lessons 10–14 cover recurrence relations, superannuation, loan mathematics, repayment calculations, and extra repayment strategies. This checkpoint tests your ability to model financial situations iteratively, calculate repayments, and compare debt-reduction strategies. Aim for 80%+ before moving to Inquiry Question 4.
8 multiple-choice questions, graded for your checkpoint score with immediate feedback. Then complete 3 written responses; these are saved for you to self-review against model answers and are not included in the MC score.
Suggested time: 35 minutes. Show formula substitutions in written responses. Keep full precision through calculations and round currency once, at the end, to the nearest cent unless a prompt says otherwise. State whether payments occur at period start or end.
Write your working, then open the matching model answer to self-check.
An investment begins with AUD 4,000 and earns 0.6% per period. A AUD 150 contribution is added at each period end. Write the recurrence and calculate A₁ and A₂.
A superannuation account starts at AUD 80,000. In this simplified model it earns 7% gross, has a 1% fee deducted from the starting balance each year, and receives AUD 9,200 at each year end for 30 years. Calculate the net rate and model the balance after 30 years.
A AUD 300,000 loan has a monthly rate of 0.5% and 240 end-of-month repayments. Calculate the repayment to cents and the total interest using the unrounded formula payment.
Question: An investment earns periodic rate r and receives deposit a at the end of each period. Which recurrence models its balance? Key: A. Interest is earned on the existing balance, then the end-of-period deposit is added: Aₙ₊₁=(1+r)Aₙ+a.
Question: A fund starts with AUD 5,000, earns 0.5% per period, and receives AUD 200 at each period end. What is the balance after one period? Key: B. A₁=1.005×5,000+200=AUD 5,225.
Question: A fund earns 7% gross annually and has an annual fee of 1.2% of the starting balance. In this simplified model, what net rate is used? Key: C. Under the question’s stated additive-fee model, net rate=7.0%−1.2%=5.8%.
Question: A loan balance accrues periodic interest r, then an end-of-period repayment M is made. Which recurrence models the balance? Key: D. Interest first changes Aₙ to (1+r)Aₙ; subtract the repayment M afterwards.
Question: A AUD 300,000 loan is repaid monthly over 20 years at 6% p.a. nominal, compounded monthly. What is the level repayment, to the nearest cent? Key: B. r=0.06/12=0.005 and n=240. M=Pr/[1−(1+r)^−n]=AUD 2,149.29. Payments are at month end.
Question: For a loan with a fixed repayment, what is the immediate mathematical effect of placing money in an offset account, in the simplified model studied? Key: C. An offset reduces the net balance used to calculate interest. The loan’s stated rate and repayment schedule are separate quantities.
Question: Why does an additional repayment early in a reducing-balance loan usually reduce total interest? Key: D. After the extra payment reduces principal, later interest accrues on a smaller balance. The effect compounds over the remaining periods.
Question: A AUD 250,000 loan accrues 0.5% interest monthly and is repaid by AUD 1,800 at each month end. The continuous formula estimate for payoff time is closest to: Key: A. n=−ln(1−Pr/M)/ln(1+r)=−ln(1−250000×0.005/1800)/ln(1.005)≈237.72 months. A scheduled recurrence is needed for the exact final payment month.
$A_{n+1}=1.006A_n+150$. $A_1=1.006(4{,}000)+150=\$4{,}174$; $A_2=1.006(4{,}174)+150=\$4{,}349.04$.
$r_{\text{net}}=7\%-1\%=6\%$. $A_{30}=80{,}000(1.06)^{30}+9{,}200\frac{(1.06)^{30}-1}{0.06}=\$1{,}186{,}814.61$. Round only the final currency value.
$M=\frac{300{,}000(0.005)}{1-(1.005)^{-240}}=\$2{,}149.29$ per month. Using the unrounded payment, total interest is $240M-300{,}000=\$215{,}830.36$. A schedule using rounded monthly payments can differ slightly.
Tick when you have finished all questions and reviewed your answers.