Loans and investments
Analyse reducing-balance loans using recurrence relations and geometric series; calculate loan repayments and outstanding balances; compare investment and loan scenarios.
Worked examples
Building a recurrence table
Straightforward
Problem
A loan of 600. Build a repayment table for the first 3 months and find the outstanding balance after 3 months.
1
Write the recurrence relation for the balance.
, with .
2
Apply the recurrence month by month.
Month 1: opening balance , interest (1%) , repayment , closing balance . Month 2: opening balance , interest , repayment , closing balance . Month 3: opening balance , interest , repayment , closing balance .
3
Interpret the pattern.
Each month a slightly larger proportion of the repayment reduces the principal, because the interest charge falls as the balance falls. This is the hallmark of a reducing-balance loan.
Answer
The outstanding balance after 3 months is .
Finding the repayment and total interest
Moderate
Problem
A borrower takes out a $25,000 car loan at 6% per annum (monthly compounding) to be repaid over 4 years. Find (a) the monthly repayment, and (b) the total interest paid.
1
Identify parameters.
Monthly rate: , months, .
2
Apply the repayment formula .
3
Find the total interest paid.
Answer
The monthly repayment is and the total interest paid is .
Deriving and using the balance formula
Challenging
Problem
Derive the formula for the outstanding balance after repayments, then use it to find the balance after 2 years (24 months) on the loan from Example 2 ( at 6% p.a. monthly, ).
1
Derive the balance formula starting from and .
The bracket is a geometric series, giving:
2
Apply the formula to the loan from Example 2 (, , , ).
3
Interpret the result.
After 2 years, roughly half the principal ($25,000) remains — reflecting the back-loading of interest in the early repayments.
Answer
, and after 24 months the outstanding balance is approximately .
Practise
Q1·Straightforward
A loan of $20{,}000 charges interest at 0.5\% per month. At the end of each month a repayment of $400 is made. The balance satisfies , with . Find , the outstanding balance after the first repayment.
Explanation
Of the $400 repayment, covered interest and $300 reduced the principal.
Q2·Straightforward
A loan of $5000 charges 1\% interest per month. Monthly repayments are $200. Starting from , apply the recurrence to find , the balance after 3 months. Round to the nearest cent.
Explanation
Q3·Straightforward
Find the monthly repayment required to repay a loan of $12{,}000 over 12 months at an interest rate of 1\% per month. Use . Round to the nearest cent.
Explanation
Q4·Moderate
The outstanding balance after repayments on a reducing-balance loan is given by
A loan of $5000 at 2\% per month has monthly repayments of $300. Find the outstanding balance after 3 months. Round to the nearest cent.
A loan of $5000 at 2\% per month has monthly repayments of $300. Find the outstanding balance after 3 months. Round to the nearest cent.
Explanation
Q5·Moderate
A loan of $10{,}000 at 1\% per month is repaid over 12 months at $888.49 per month. Calculate the total interest paid over the life of the loan. Round to the nearest cent.
Explanation
Total repayments
Total interest
Note: this relatively small interest amount reflects the short loan term. Over longer terms the total interest grows substantially.
Total interest
Note: this relatively small interest amount reflects the short loan term. Over longer terms the total interest grows substantially.
Q6·Moderate
Find the monthly repayment to fully repay a $30{,}000 loan over 5 years (60 months) at an interest rate of 6\% per annum, compounded monthly. Round to the nearest cent.
Explanation
Monthly rate: ,
Q7·Moderate
Loan A: $20{,}000 at 6\% per annum (monthly compounding) repaid over 3 years.
Loan B: $20{,}000 at 5\% per annum (monthly compounding) repaid over 4 years.
Calculate the total interest paid on Loan A. Round to the nearest cent.
Loan B: $20{,}000 at 5\% per annum (monthly compounding) repaid over 4 years.
Calculate the total interest paid on Loan A. Round to the nearest cent.
Explanation
Monthly rate: ,
Total paid:
Total interest:
Total paid:
Total interest:
Q8·Moderate
For the loans from Q7:
- Loan A: $20{,}000 at 6\% p.a. (monthly) over 3 years — total interest $1903.79
- Loan B: $20{,}000 at 5\% p.a. (monthly) over 4 years
Calculate the total interest paid on Loan B. Round to the nearest cent.
- Loan A: $20{,}000 at 6\% p.a. (monthly) over 3 years — total interest $1903.79
- Loan B: $20{,}000 at 5\% p.a. (monthly) over 4 years
Calculate the total interest paid on Loan B. Round to the nearest cent.
Explanation
Monthly rate: ,
Total paid:
Total interest:
Loan A costs less in total interest ($1903.79 vs $2108.12) even though its rate is higher, because the shorter term limits interest accumulation.
Total paid:
Total interest:
Loan A costs less in total interest ($1903.79 vs $2108.12) even though its rate is higher, because the shorter term limits interest accumulation.
Q9·Challenging
A reducing-balance loan satisfies the recurrence , with . By recognising the geometric series that arises when the recurrence is expanded, show that
Then use this formula to find the outstanding balance after 24 months on a $30{,}000 loan at 0.5\% per month with monthly repayment $333.06. Round to the nearest cent.
Then use this formula to find the outstanding balance after 24 months on a $30{,}000 loan at 0.5\% per month with monthly repayment $333.06. Round to the nearest cent.
Explanation
**Deriving the formula:**
Expanding the recurrence:
The bracket is a geometric series with terms, first term 1, ratio :
**Numeric calculation** (, , , ):
Expanding the recurrence:
The bracket is a geometric series with terms, first term 1, ratio :
**Numeric calculation** (, , , ):
Q10·Challenging
A loan of $8000 at 1\% monthly interest is repaid in monthly instalments of $500. After how many full months will the loan first be fully repaid?
Explanation
Setting :
Let :
So the smallest integer is .
Verification: ; . ✓
Let :
So the smallest integer is .
Verification: ; . ✓
Q11·Challenging
A borrower takes out a $200{,}000 home loan at 6\% per annum compounded monthly over 25 years. The monthly repayment is $1288.60. After 10 years (120 months) of repayments, the borrower makes a one-off extra payment of $20{,}000 off the principal.
How many months does this shorten the loan by?
How many months does this shorten the loan by?
Explanation
**Balance after 120 months** (, ):
**After extra payment:**
**Months to repay reduced balance** at $1288.60:
**Original remaining months:**
**Months saved:** months.
**After extra payment:**
**Months to repay reduced balance** at $1288.60:
**Original remaining months:**
**Months saved:** months.
Q12·Challenging
An investor deposits $500 per month into a managed fund returning 5\% per annum (compounded monthly) for 10 years.
A friend takes out a $50{,}000 personal loan at 8\% per annum compounded monthly and repays it over 10 years.
Find the difference between the investor's fund value and the total interest paid by the friend. Round to the nearest dollar.
A friend takes out a $50{,}000 personal loan at 8\% per annum compounded monthly and repays it over 10 years.
Find the difference between the investor's fund value and the total interest paid by the friend. Round to the nearest dollar.
Explanation
**Investor's fund** (, ):
**Friend's loan** (, , ):
Total interest
**Difference:**
The investor ends the decade $54{,}845 better off in fund value than the friend paid in interest.
**Friend's loan** (, , ):
Total interest
**Difference:**
The investor ends the decade $54{,}845 better off in fund value than the friend paid in interest.
Open Math
Loans and investments
Algebra / Functions · MAV-12-08
Name:
Date:
Q1Straightforward
A loan of $20{,}000 charges interest at 0.5\% per month. At the end of each month a repayment of $400 is made. The balance satisfies , with . Find , the outstanding balance after the first repayment.
Q2Straightforward
A loan of $5000 charges 1\% interest per month. Monthly repayments are $200. Starting from , apply the recurrence to find , the balance after 3 months. Round to the nearest cent.
Q3Straightforward
Find the monthly repayment required to repay a loan of $12{,}000 over 12 months at an interest rate of 1\% per month. Use . Round to the nearest cent.
Q4Moderate
The outstanding balance after repayments on a reducing-balance loan is given by
A loan of $5000 at 2\% per month has monthly repayments of $300. Find the outstanding balance after 3 months. Round to the nearest cent.
A loan of $5000 at 2\% per month has monthly repayments of $300. Find the outstanding balance after 3 months. Round to the nearest cent.
Q5Moderate
A loan of $10{,}000 at 1\% per month is repaid over 12 months at $888.49 per month. Calculate the total interest paid over the life of the loan. Round to the nearest cent.
Q6Moderate
Find the monthly repayment to fully repay a $30{,}000 loan over 5 years (60 months) at an interest rate of 6\% per annum, compounded monthly. Round to the nearest cent.
Q7Moderate
Loan A: $20{,}000 at 6\% per annum (monthly compounding) repaid over 3 years.
Loan B: $20{,}000 at 5\% per annum (monthly compounding) repaid over 4 years.
Calculate the total interest paid on Loan A. Round to the nearest cent.
Loan B: $20{,}000 at 5\% per annum (monthly compounding) repaid over 4 years.
Calculate the total interest paid on Loan A. Round to the nearest cent.
Q8Moderate
For the loans from Q7:
- Loan A: $20{,}000 at 6\% p.a. (monthly) over 3 years — total interest $1903.79
- Loan B: $20{,}000 at 5\% p.a. (monthly) over 4 years
Calculate the total interest paid on Loan B. Round to the nearest cent.
- Loan A: $20{,}000 at 6\% p.a. (monthly) over 3 years — total interest $1903.79
- Loan B: $20{,}000 at 5\% p.a. (monthly) over 4 years
Calculate the total interest paid on Loan B. Round to the nearest cent.
Q9Challenging
A reducing-balance loan satisfies the recurrence , with . By recognising the geometric series that arises when the recurrence is expanded, show that
Then use this formula to find the outstanding balance after 24 months on a $30{,}000 loan at 0.5\% per month with monthly repayment $333.06. Round to the nearest cent.
Then use this formula to find the outstanding balance after 24 months on a $30{,}000 loan at 0.5\% per month with monthly repayment $333.06. Round to the nearest cent.
Q10Challenging
A loan of $8000 at 1\% monthly interest is repaid in monthly instalments of $500. After how many full months will the loan first be fully repaid?
Q11Challenging
A borrower takes out a $200{,}000 home loan at 6\% per annum compounded monthly over 25 years. The monthly repayment is $1288.60. After 10 years (120 months) of repayments, the borrower makes a one-off extra payment of $20{,}000 off the principal.
How many months does this shorten the loan by?
How many months does this shorten the loan by?
Q12Challenging
An investor deposits $500 per month into a managed fund returning 5\% per annum (compounded monthly) for 10 years.
A friend takes out a $50{,}000 personal loan at 8\% per annum compounded monthly and repays it over 10 years.
Find the difference between the investor's fund value and the total interest paid by the friend. Round to the nearest dollar.
A friend takes out a $50{,}000 personal loan at 8\% per annum compounded monthly and repays it over 10 years.
Find the difference between the investor's fund value and the total interest paid by the friend. Round to the nearest dollar.
Worked solutions and answers at openmath.au/year-12/advanced/financial-mathematics/loans-and-investments