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NSWMaths AdvancedQuick questions
Year 12: Financial Mathematics
Quick questions on Reducing-balance loans: repayments, outstanding balance and present value of an annuity
14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is the recurrence model?Show answer
A loan of $P$ is repaid by equal payments $M$ at the end of each period. Interest of rate $r$ per period accrues on the outstanding balance. Let $B_n$ be the balance just after the $n$th payment, with $B_0 = P$.
What is the repayment formula (present value of an annuity)?Show answer
The loan is fully repaid when $B_n = 0$. Setting the closed form to zero and solving for $M$:
What is splitting a payment into interest and principal?Show answer
The interest portion of the $k$th payment is the interest charged on the previous balance:
What is total interest?Show answer
Total amount paid over the loan is $M n$. Since the loan amount $P$ is returned, total interest is
What is time to repay?Show answer
If a borrower chooses $M$ instead of $n$, the loan term is
What is standard repayment?Show answer
$\$25000$ car loan at $7.2\%$ per annum compounded monthly, repaid over $5$ years.
What is outstanding balance partway through?Show answer
For the loan above, balance after $24$ payments:
What is interest vs principal in one payment?Show answer
For the same loan, the $25$th payment is split as
What is solving for the term?Show answer
You can afford $\$1500$ a month on a $\$300000$ loan at $6\%$ per annum compounded monthly. How many months will it take?
What is using the future-value formula for a loan?Show answer
Loan repayments use the present-value-of-annuity formula. The future-value formula is for accumulating savings.
What is forgetting to convert the rate?Show answer
Monthly compounding requires the monthly rate $R / 12$. Using the annual rate gives a wildly wrong answer.
What is mixing up the two forms?Show answer
$M = \frac{P r (1 + r)^n}{(1 + r)^n - 1}$ and $M = \frac{P r}{1 - (1 + r)^{-n}}$ are the same. Pick one and use it consistently.
What is treating principal repaid as $M \cdot$ fraction?Show answer
Early payments are mostly interest; the principal-repaid pattern is geometric, not linear. Use $P - B_n$ for the amount of principal repaid after $n$ payments.
What is ignoring the no-completion case?Show answer
If $M \le P r$, the loan never finishes. The log formula will fail or give a negative result.